n

A COMPARISON OF THREE TYPES OF ITEM ANALYSISIN TEST DEVELOPMENT USING CLASSICAL

AND LATENT TRAIT METHODS

By

IRIS G. BENSON

A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OFTHE UNIVERSITY OF FLORIDA

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THEDEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

1977

^iiiiii

ACKNOWLEDGMENTS

I am deeply indebted to two special people who have greatly in-

fluenced by graduate education, Dr. William Ware, chairman of my

guidance committee, and Dr. Linda Crocker, unofficial cochairman of my

committee. Their continued encouragement and support has resulted in

my reaching this point in my graduate studies. I shall always be

extremely grateful to Dr. Ware and Dr. Crocker for whatever skills I

have developed as a researcher and as a teacher are in large part due

to their advice and guidance. To them 1 owe the high value I place on

objective, quantitative research methods. Further, 1 would like to

acknowledge the tremendous amount of time they spent in molding the

final copy of this manuscript.

1 would also like to express my appreciation to the members of my

committee. Dean John Newell and Dr. William Powell, for their suggestions

and editorial coimnents on this dissertation. Special thanks are extended

to Dr. Wilson Guertin for his assistance with portions of the study, and

as an unofficial member of my committee.

1 would like to thank Dr. Jeaninne Webb, Director of the Office of

Instructional Resources, and Mr. Robert Feinberg and Ms. Arlene Barry

of the Testing Division, for providing the data used in this study.

Finally, I would like to express my sincere appreciation to my

friends and family who stood by me during very trying times in my

graduate education.

11

y

TABLE OF CONTENTS

PAGE

ACKNOWLEDGMENTS ii

LIST OF TABLES y

LIST OF FIGURES vii

ABSTRACT ^iii

CHAPTER

I. INTE^ODUCTION1

The Problem 7

Purpose of the Study 8

Significance of the Study 10Organization of the Study H

II. REVIEW OF THE LITERATURE 13

Item Analysis Procedures for the Classical Model .... 13Research Related to Classical Item Analysis in

Test Development 18Simplified Methods of Obtaining ItemDiscrimination 21

Item Analysis Procedures for the Factor AnalyticModel 22Research Related to Factor Analysis in Test

Development .. , 24Comparison of .Factor Analysis to^-CTassical Item

Analysis . . "/'"r'""!^ T'*". . . . 7"""7""T' 25Item Analysis Procedures for the Latent Trait Model ... 27Research Related to Latent Trait Models in

Test Development 32Comparison of the fesch Model to Factor Analysis ... 34

Summary . .

'.'. .

"7~": :"". 36

III. METHOD 39

The Sample 39The Instrument 40The Procedure 42

Design 42Item Selection 44Double Cross-Validation j:_„---.-^—

-

•-—

'

-•—;. • • • 46Statistical Analyses , T'T 48

Summary 51

111

TABLE OF CONTENTS - Continued

CHAPTER PAGE

IV. RESULTS 53

Item Selection 54Double Cross-Validation 69Comparison of the 15 Item Tests on Precision 69Comparison of the 30 Item Tests on Precision 75Comparison of the 30 Item Tests on Lfficienc)' 81Summary 82

V. DISCUSSION AND CONCLUSIONS 87

The Precision of the Tests Produced by theThree Methods of Item Analysis 87Internal Consistency 88Standard Error of Measurement 89Types of Items Retained 90Conclusions 9I

The Efficiency of the Tests Produced by theThree Methods of Item Analysis 93Conclusions 95

Implications for Future Research 95

VI. SUMMARY 99

REFERENCES IO5

APPENDIX A: Mathematical Derivation of the Rasch Model .... 112

APPENDIX B: Relative Efficiency Values Used in Figure2 for the Comparisons Among Item AnalyticMethods 119

BIOGRAPHICAL SKETCH 120

IV

LIST OH TABLES

TABLE PAGE

1 DESCRIPTIVE DATA ON THE VERBAL APTITUDE SUBTEST

OF THE FLORIDA TWELFTH GRADE TEST 1975

ADMINISTRATION 41

2 SYSTEMATIC SAMPLING DESIGN OF THE STUDY N - 5,235 ... 43

3 DOUBLE CROSS-VALIDATION DESIGN OF THE STUDY 47

4 DEMOGRAPHIC BREAKDOWN BY ETHNIC ORIGIN AND

SEX FOR TOTAL SAMPLE 55

5 SUMMARY STATISTICS ON THE 50 TEST ITEMS BASED

ON CLASSICAL ITEM ANALYSIS FOR EACH SAMPLE SIZE .... 56

6 ITEM LOADINGS ON THE FIRST UNROTATED FACTOR FOR

THE 50 TEST ITEMS BASED ON FACTOR ANALYSIS FOR

EACH SAMPLE SIZE 58

7 SUMMARY STATISTICS ON THE 50 TEST ITEMS BASED

ON THE RASCH MODEL FOR EACH SAMPLE SIZE 61

8 DESCRIPTIVE DATA ON ITEM DISCRIMINATION ESTIMATES

BASED ON THE RASCU MODEL ACCORDING TO SAMPLE SIZE ... 65

9 THE 15 BEST ITEMS SELECTED UNDER EACH ITEM ANALYTIC

PROCEDURE ACCORDING TO SAMPLE SIZE 66

10 THE 30 BEST ITEMS SELECTED UNDER EACH ITEM ANALYTIC

PROCEDURE ACCORDING TO SAMPLE SIZE 67

11 DESCRIPTIVE STATISTICS FOR THE TEST COMPOSED OF

THE 15 BEST ITEMS SELECTED BY EACH ITEM

ANALYTIC PROCEDURE ACCORDING TO SAMPLE SIZE 70

12 CONFIDENCE INTERVALS FOR THE OBSERVED INTERNAL

CONSISTENCY ESTIMATES BASED ON THE 15 ITEM

TESTS ACCORDING TO SAMPLE SIZE 72

13 15 ITEM TESTS: DESCRIPTIVE STATISTICS FOR ITEM

DIFFICULTY BY PROCEDURE AND SAMPLE SIZE 74

14 15 ITEM TESTS: DESCRIPTIVE STATISTICS FOR ITEM

DISCRIMINATIONS BY PROCEDURE AND SAMPLE SIZE 75

15 POST HOC COMPARISONS OF THE DIFFERENCES BETWEEN

THE MEAN ITEM DISCRIMINATIONS FOR THE 15 ITEM TESTS . . 76

V

LIST OF TABLES - Continued

TABLE PAGE

16 DESCRIPTIVE STATISTICS 1-OR THE TEST COMPOSEDOF THE 30 BEST ITEMS SELECTED BY EACH ITEMANALYTIC PROCEDURE ACCORDING TO SAMPLE SIZE 77

17 CONFIDENCE INTERVALS FOR THE OBSERVEDINTERNAL CONSISTENCY ESTIMATES BASED ONTHE 30 ITEM TESTS ACCORDING TO SAMPLE SIZE 78

18 30 ITEM TESTS: DESCRIPTIVE STATISTICS FORITEM DIFFICULTY BY PROCEDURE AND SAMPLESIZE 80

19 50 ITEM TESTS: DESCRIPTIVE STATISTICS FORITEM DISCRIMINATIONS BY PROCEDURE AND SAMPLESIZE 80

VI

LIST OF FIGURES

FIGURE PAGE

1 HYPOTHETICAL ITEM CHARACTERISTIC CURVES FORTHE FOUR LATENT TRAIT MODELS 29

2 RELATIVE EFFICIENCY COMPARISONS FOR THETHREE 30 ITEM TESTS N = 995 S3

Vll

Abstract of Dissertation Presented to tl\e GraduateCouncil of the University of Florida in Partial Fulfillmentof tlie Requirements for the Degree of Doctor of P'ailosophy

A COMPARISON OF THREE TYPES OF ITEM ANALYSISIN TEST DEVELOPMENT USING CLASSICAL

AND LATENT TRAIT METHODS

By

Iris G. Benson

December 1977

Clia i rman : Willi am B . Wa r e

Major Department: Foundations of Education

Test reliability and validity are determined by the quality of the

items in the tests. Tlirough the application of item analysis procedures,

test constructors are able to obtain quantitative, objective information

useful in developing and judging the quality of a test and its items.

Classical test theory forms the basis for one method of test

development. An integral part of the development of tests based on the

classical model is selection of a final set of items from an item pool

based on classical item analysis or factor analysis. Classical item

analysis requires identification of single items which provide maximum

discrimination between individuals on the latent trait being measured.

The bi serial correlation between item score and total score is coimnonly

used as an index of item discrimination.

An alternative method of test development, but based on the

classical model, is factor analysis. Factor analysis is a more complex

test development procedure than classical item analysis. It is a

VI 11

statistical teclmique that takes into account the item correlation

with all other individual items in the test simultaneously. Thus,

classical item analysis can be viewed as a unidimensional basis for

item analysis, less sophisticated tlian tlie multidimensional procedure

of factor analysis.

Recently, the field of latent trait theory has provided a new approach

to test construction. Several latent trait models have been developed;

however, this study was concerned only with the one-parameter logistic

Rasch model. The Rasch model was chosen because it is the most parsi-

monious of the latent trait models and has recently been used in the

development and equating of tests.

A review of the literature revealed numerous studies conducted in

each of the three areas of item analysis, but no comparative studies

were reported among all three item analytic techniques. Therefore,

the present study was designed to compare the methods of classical

item analysis, factor analysis, and the Rasch model in terms of test

precision and relative efficiency.

An empirical study was designed to compare the effects of the

three methods of item analysis on test development across different

sample sizes of 250, 500, and 995 subjects. Item response data were

obtained from a sample of 5,235 high school seniors on a 50 item cogni-

tive test of verbal aptitude. The subjects were divided into nine

independent samples, one for each item analytic technique and sample

size. The study was conducted in three phases: item selection,

computation of item and test statistics for selected items on double

cross-validation samples, and statistical analyses of item characteris-

tics. For each item analytic procedure two tests were developed:

IX

a 15 item test, and a 50 item test. Four dependent variables were

obtained for eacli test to assess precision: internal consistency

estimates, standard error of measurement, item difficulties, and item

discriminations. In addition, tlic relative efficiencies of the 30 item

tests developed by each item analytic technicjue were compared for the

sample of 995 subjects.

The results of the analysis revealed tliat there were no differences

between the tests developed by the three methods of item analysis,

in terms of the precision of measurement. In terms of efficiency,

substantive differences between the tests produced by the three item

analytic methods were observed. Specifically, the tests based on class-

ical test theory were more effective for measuring very low and very

high ability students. Tlie Rasch developed test was more efficient for

assessing average and high ability students.

CHAPTFR I

INTRODUCTION

The systematic approach to test development was initiated by Binet

and Simon in 191(i. Since that time psychometricians have been concerned

with the extent to wliich accurate measurement of a person's "ability"

is possible. Most measurement experts agree that upon repeated testing

an individual's observed score will vary even though his true ability

remains constant. Tliis variability is the essence of classical test

tlieory

.

Classical test theory is based upon the assumption that a person's

observed score (X) is made up of a true score (T) and error score (E)

denoted:

X = T + E. (1)

Limited by few assumptions, this theory has wide applications. The few

assumptions pertain to the eri'or score (Magnusson, 1966, p. 64):

1. Tlie mean of an examinee's error scores on an infinite

number of jiarallel tests is zero.

2. The correlation between examinee's error scores on parallel

tests is zero.

3. The correlation between examinees' error scores and true

scores is zero.

Relying upon these assumptions, psychometricians liave used the observed

score (X) to represent the best estimate of a person's true score (T)

.

1

The accuracy of the observed score [XJ in representing an examinee's

true score (T) is described by the reliability coefficient. One definition

of reliability is given by the coefficient of precision. This coefficient

is the correlation between truly parallel tests, assuining the examinee's

true score does not change between two measurements. Lord and Novick

(1968] have defined truly parallel tests to be those for which, "the

expected values [true scores] of parallel measurements are equal; and the

observed score variances of parallel measurements are equal (p. 4S)."

The reliability coefficient for the population is defined as

(Lord and Novick, 1968, p. 134):

2 ''

r = "^T^ - 1 - ^e" , (2)XX ^- —

2 2where o is the true score variance, o,, is the observed score variance,

i A

and Op is the error score variance. IVhen this expression is used to

represent the coefficient of precision, it can be interpreted as the

extent to which unreliability is due solely to inadequacies of the test

form and testing procedure rather than due to changes in examinees over

time.

The coefficient of precision is a theoretical value because the

2 2components o and o cannot be observed. The coefficient of precision

is usually estimated by internal consistency methods. Internal con-

sistency is a measure of the relationship between random parallel tests.

Random parallel tests are composed of items drawn from the same population

of items (Magnusson, 1966, p. 102-105). Scores on these tests may

differ somewhat from true scores in means, standard deviations, and

correlations because of random errors in the sampling of items. However,

random parallel tests are more often encountered in practice than are

truly parallel tests. Cronbach's coefficient alpha (1951) is the

internal consistency coefficient commonly used to represent tlie average

correlation among all possible tests created by dividing the domain

into random lialves. Thus, the intern.al consistency coefficient indicates

the extent to which all tlie items are measuring the same ability or trait,

Psychological traits are often described as latent because they cannot

be directly observed. Therefore, psychological tests are developed in

an attempt to measure these latent traits.

Classical test tlieory forms the basis for one metliod of test

development. An integral part of the development of tests based on the

classical model is the utilization of classical item analysis or factor

analysis. Classical item analysis is a procedure to obtain a description

of the statistical cliaracteristics of each item in the test. This

approach requires identification of single items which provide maximum

discrimination between individuals on the latent trait being measured.

Theoretically, selecting items which have high correlations with total

test score will result in a discriminating test winch is homogeneous

with respect to tlie latent trait. Therefore, classical item analysis

is an aid to developing internally consistent tests.

An alternative method of test development, but based on the classi-

cal model, is factor analysis. Factor analysis is a more complex test

development procedure than classical item analysis. It is a statistical

technique that takes into account the item correlation with all other

individual items in the test simultaneously. Groups of similar items

tend to cluster together and comprise the latent traits (factors) under-

lying the test. Under the classical model then, classical item analysis

can be viewed as a unidimensional basis for item analysis, less

sophisi ticated than the multidimensional ]irocedure of factor analysis.

The purpose of factor analysis is to represent a variable in terms

of one or several underlying factors lHarman, 1967). Depending upon the

objective of the analysis, two general approaches are used in

factor analysis: (a) common factor analysis, and (h) principal com-

ponents analysis. A common factor solution would he warranted if the

researcher were interested in determining the number of common and

unique factors underlying a given test. A principal component solution

would be warranted if it were of interest to extract the maximum

amount of variance from a given test.

Regardless of the apju-oach used, factor analysis is an item analytic

technique in whicli all test items are considered simultaneously to pro-

duce a matrix of item correlations with factors. It is these correlations

or item loadings tliat indicate the strength of the factor and also the

number of factors underlying the test. However, factor analysis shares

the weakness of classical item analysis, that of being sample dependent.

Critics of classical test theory contend that a major weakness of

tests developed from this model is that the item statistics vary when

the examinee group changes; item statistics may also vary if a different

set of items from the same domain is used with the same examinee group

(llambleton and Cook, 1977; Wright, 1968). Thus, the selection of a final

set of test items will be sample dependent.

Until recently, classical item analysis and factor analysis were tlie

only techniques described in measurement texts for use in item analysis

and test development (Baker, 1977). However, with the publication of

Lord and Novick's Statistical Theories of Mental Test Scores (1968) and

the availability of computer programs, considerable attention is being

directed now toward the field of latent trait tlieory as a new area in

test development. Latent trait theory dates back to Lazarsfeld (1950)

who introduced the concept; however, Fredrick Lord is generally given

credit as the father of latent trait theory (Hambleton, Swaminathan,

Cook, Eignor, and Gifford, 1977). Proponents of this approach claim

that the advantages of latent trait theory over classical test theory

are twofold: (a) theoretically it provides item parameters which are

invariant across examinee samples which will differ with respect to

the latent trait, and (b) it provides item characteristic curves that

give insight into how specific items discriminate between students of

varying abilities. These properties of latent trait theory will be

presented in more detail in Chapter II.

Four latent trait models have been developed for use with

dichotomously scored data: the normal ogive, and tlie one-, two-, and

three-parameter logistic model (Hambleton and Cook, 1977; Lord and Novick,

1968). This study is concerned with the one-parameter logistic Rasch

model because it is the simplest of the four models.

Tests developed using the Rasch model are intended to provide

objective measurement of the examinee's true ability on the latent

trait in question, as well as providing for invariant item parameters

(Rasch, 1966; Wright, 1968). lliat is, any subset of items from a

population of items that have been calibrated by the Rasch model should

accurately measure the examinee's true ability regardless of whether the

items are very easy or very difficult; also, the item parameters should

remain constant over different examinees. In measurements obtained from

classical test theory this objective feature is rarely attained. The

item parameters associated with classical test theory are group and

item specific. That is, the item parameters are determined by the

ability of the people taking the test and the subset of items chosen.

Wright (196S) has stated, "llie growth of science depends on the develop-

ment of objective methods for transforming an observation into measure-

ment (p. S6)." Latent trait theory is an attempt to develop mental

measurement into a technique similar to measurement in the physical

sciences

.

Latent trait theory is based on strong assumptions that are re-

strictive and lience limit its application [Hambleton and Cook, 1977). The

assumptions required for the Rasch model are the following (Rasch, 1966):

1. Tlie test is unidimensional, e.g., there is only one factor

or trait underlying test performance.

2. The item responses of each examinee are locally independent,

e.g., success or failure on one item does not hinder other item responses.

3. Tlie item discriminations are equal, e.g., all items load

equally on the factor underlying the test.

Lord and Xovick (1968) noted that the assumptions of unidimensionality

and local independence are synonymous. To say that only one underlying

ability is being tested means the items are statistically independent

for persons at the same ability level. The third assumption relates to

item characteristic curves. Tlie item characteristic curve is a mathematical

function that relates the probiibility of success on an item to the

ability measured by tlie test. Curves vary in slope and intercept to

reflect how items vary in discrimination and difficulty. Tlie one-param-

eter logistic Rasch model (the one parameter is item difficulty) assumes

all item discriminations are equal. Tlius all item characteristic

curves should be similai- with respect to their slopes.

The Problem

Several studies have been conducted to varify the invariant prop-

erties of tests constructed using the Rascli model (Tinsley and Dawis,

1975; IVhitely and Dawis, 1974; Wright, 1968). If we assume that tests

developed using latent trait theory possess the quality of invariant

item statistics, why then liasn't latent trait theory been more visible

in the psychometric community? There appear to be three main reasons

for this slow acceptance. First, the Rasch procedure is based on a

mathematical model involving restrictive assumptions, e.g., the uni-

dimensionality of the items, the local independence of the items, and

equal item discriminations. A further restriction of the Rasch model

is the assumption of minimal guessing. However, several researchers

have demonstrated the robustness of the model with regard to departures

from the basic assumptions (Anderson, Kearney and Everett, 196S; Dinero

and Haertel, 1976; Rentz, 1976). Second, latent trait theory has not been

used in practical testing situations because until recently there was a

lack of available computer programs to handle the complex mathematical

calculations. Hambleton et al. (1977) described four computer programs

now available to the consumer. Third, measurement experts who are

knowledgeable about latent trait models have been skeptical as to tlie

real gains that may be available through this line of research. Are

tests developed using latent trait models superior to tests developed

using classical item analysis or factor analysis?

The purpose of this study was to compare the precision and efficiency

of cognitive tests constructed by tlie three methods (classical item

analysis, factor analysis and the Rasch model) from a common item and

examinee population. Precision, as measured by internal consistency,

8

is an overall estimate of a test's homogeneity, but provides no infor-

mation on how the test as a whole discriminates for tlie various ability

groups taking tlie test. For that reason measures test efficiency

(Lord, 1974a, 1974b) were incorporated into the study. Test efficiency

provides information on tlie effectiveness of one test over another as

a function of ability level. A cognitive college admissions subtest was

used in this study for several reasons. First, tests of this t)^e are

widely used by educational institutions for a large number of examinees

each year, in the areas of selection, placement, and academic counseling.

Most college admission examinations traditionally have been developed

using classical item analysis. Second, because of the importance of

the decisions made using such test scores, it would be worth investing

considerable time and expense in the development of these instruments.

Thus, the use of factor analysis or the Rascli model would be justified

if superiority of either of these methods over classical item analysis

could be determined. Third, tlie items on college admission tests have

been written by experts, and each subtest is intended to be unidimensional

,

e.g., items measuring a single ability. Thus, assumptions from all

models should be met. Fourth, because of the time required to take such

examinations, it is important to maximize the precision and tlie effect-

iveness of tlie tests. Tlie possibility of using fewer items while main-

taining precision would be desirable. Therefore, the question of which

test development procedure can best accomplisli this is not a trival one.

Purpose of the Study

The purpose of this study was to compare empirically the Rasch model

with classical item analysis and factor analysis in test development.

Five research questions guided tiiis study.

1. Kill the three methods of test development produce tests with

superior internal consistency estimates when compared to the jirojected

internal consistency of the population as the number of items decreases?

2. Will the three methods of test development produce tests

with stable estimates of internal consistency when the number of

examinees decreases?

3. Will the tliree methods of test development produce tests

with similar standard errors of measurement?

4. Will the three methods of test development select items that

are similar in terms of difficulty and discrimination?

5. Will the three methods of test development produce equally

efficient tests for all ability levels?

Hypotheses

This study investigated the capacities of three methods of test

development to increase precision and efficiency of measurement in test

construction. Tlie five questions posited in the previous section were

phrased as testable hypotheses:

1. There are no significant differences in the internal consistency

estimates of the tests produced by the three methods, as the number of items

decreases, wlien compared to the projected internal consistency estimates

for the population for tests of similar length.

The standard error of measurement (SEM) is defined in the classicalsense as (Magnusson, 1966, p. 79):

SEM = Sv vT~~rXX,

where Sv is the standard deviation of the test, and r is theXX

reliability coefficient.

10

2. There are no differences in the internal consistency estimates

of the tests produced by the three methods when the number of examinees

is decreased.

3. There are no meaningful" differences in the magnitude of the

standard error of measurement of the tests produced by the three

methods

.

4. There are no significant differences in the difficulties or

discriminations of the items selected by the three methods.

5. There are no differences across ability levels in the efficiency

of the tests produced by the three methods.

Significance of the Study

Objective measurement has always been assumed in the physical

sciences. It has only been recently that objective measurement in the

behavioral sciences has been deemed possible with the advent of latent

trait theory. Since the introduction of latent trait theory by

Lazarsfeld (1950) and Lord (1952a, 1953a, 1953b) much of the research

on latent trait models has been confined to theoretical research journals,

Wright (1968), speaking at a conference on testing problems, discussed

at an applied level the need to seriously consider latent trait theory

and the Rasch model in particular as a major test development technique

far superior to classical item analysis and factor analysis. However,

even in 1968 computer programs were not yet available to run the analyses

2Because test scores are usually reported and interpreted in

whole numbers, a "meaningful" difference in the standard error ofmeasurement is defined as a difference of ii 1.00.

11

should anyone beyond academicians be intex'csted. Today this obstacle

has been overcome, but many test developers remain unconvinced of the

value of latent trait theory because its superiority to classical test

theory has not been conclusively demonstrated. This study is an attempt

to provide an empirical comparison of classical test theory and latent

trait tlieory methods of test construction.

Of the various logistic models that represent latent trait theory

the Rasch model was chosen for comparison with traditional item

analysis procedures in the present study because it is the most

parsimonious latent trait model and has been used recently in the

development of the equating of tests fRentz and Bashaw, 1977; Woodcock,

1974). Tlie Rascli model provides a matliematical explanation for the

outcome of an event when an examinee attempts an item on a test. Rasch

(1966) stated that the outcome of an encounter is governed by the pro-

duct of tlie ability of tlie examinee and the easiness of the item and

nothing more. The imi)lication of this simple concept (objectivity of

measurement) would seem to revolutionize mental measurement. If invariant

properties of items and ability scores can be identified and used to

improv'e the psychometric quality of tests to an extent greater than now

possible with classical and factor analytic procedures then \ip ti-uly

are in the age of modern test theory.

Organization of the Study

The theoretical and empirical studies related to the three methods

of item analysis are described in Chapter II. An empirical investigation

to compare tlie three metliods of item analysis under varying conditions

is described in Chapter III. The results of the study are reported in

Chapter IV. A discussion of the results, conclusions of the study, and

12

implications for future research in this area have been presented in

the fiftli chapter. A sujiimarization of the study lias been provided in

Chapter VI.

CHAPTIZR II

REVIEW OF THE LITEilATURE

The quality of the items in a test determine its validity and

reliability. Tlirough tlie application of item analysis procedures, test

constructors are able to obtain quantitative objective information

useful in judging the quality of test items. Item analysis thus pro-

vides an empirical basis for revising the test, indicating which items

can be used again and which items have to be deleted or rewritten

(Lange, Lehmann, and Mehrens , 1967). Item analysis data also help

settle arguments and objections to specific items that might be raised

by administrators, test experts, examinees, or the public.

This study is focused on three approaches to item analysis (classical

item analysis, factor analysis, and the Rasch model) as test construction

techniques. It is assumed throughout this study that the test under

construction is unidimensional , e.g., all items are measuring only one

ability. These three approaches to item analysis and the relevant

research related to each method are discussed in this chapter.

Item Analysis Procedures for the Classi cal Model

Item analysis as a test development technique emerged at the begin-

ning of this century. Binet and Simon (1916) were among the first to

systematically validate test items. They noted the proportion of

students at particular age levels passing an item. This statistic was

13

14

measuring the relative difficulty of the items for different age groups.

The item difficulty index, defined as the percentage of persons passing

an item and denoted by p, is one ' of the statistics used in classical

item analysis.

Item difficulty is related to item variance and hence to the

internal consistency of the test. Test constructors are usually con-

cerned with achieving high test reliability, e.g., precision of measure-

ment. Therefore, an item difficulty of .50 is considered to be the ideal

value necessary to maximize test reliability. Tliis is because half

the examinees are getting the item correct and half the examinees are

missing the item. The proportion missing an item is defined as 1-p or

q. Tlius, when p is equal to .50, q is equal to .50. Uecause the

variance of a dichotomized item is p x q tlie maximum variation an item

can contribute to total test variance and ultimately to true-score

variance is .25. As an item's difficulty index deviates from .50, its

contribution to total test variance is always some value less than .25.

Hence test constructors have been advised (Gulliksen, 1945) to select

items with difficulty indices at or near .50. However, when items

are presented in multiple choice or alternate choice format, the ideal

3level of difficulty is adjusted to accommodate for guessing.

A second important item statistic in classical item analysis is

the item discrimination index. An item discrimination index is a measure

•

The ideal value of p = .50 assumes there has been no guessing onthe item. Tlie effects of guessing on item difficulty tends to increasethe ideal value of p. For example, on a four option multiple choiceitem the chance of guessing the correct answer is (J4)

( .501 = . 12. Thevalue of .12 is added to .50 to correct for the effect of guessing andthe ideal p would now be .62 (Lord, 1952b; Mehrens and Lehmann, 1975).

15

of how well the item discriminates between persons who have high test

scores and persons who have low test scores. Tlie discrimination index

is often expressed as a correlation between the item and total test

score. Ulien the criterion is total test score, the correlation coef-

ficient indicates the contribution that item makes to the test as a

whole, 'riius, on tests of academic achievement it is a measure of item

validity as well as a contributor to internal consistency. Noting an

increasing use of item analytic procedures for the improvement of

objective examinations, Richardson (1936) pointed out that the

development of tlie procedures of item analysis had centered primarily

around the invention of various indices of association between the test

item and tiie total test score, e.g., item discrimination indices.

The two most popular item-test correlation indices are the biserial

and point biserial correlations. The point biserial was developed by

Pearson (1900) and is a special case of the more general Pearson Product

Moment (PPM) correlation coefficient (Magnusson, 1966). Tliis index

is recommended when one of the variables being correlated (the item

score) represents a true dichotomy and the other variable (total test

score) is continuously distributed. Pearson (1909) also derived the

biserial correlation which is an estimate of the PPM. The biserial

correlation is recoiimiended when one of the variables (the item score)

has an underlying continuous and normal distribution which has been

artifically dichotomized and the other variable (total test score) is

continuously distributed. The assumption for the point biserial

correlation is often hard to justify when it is suspected that knowledge

required to answer an item is continuously distributed.

16

In considei'ing the dichotomized item [pass/fail], McNemar (1962]

has commented, "It is obvious that failing a test item represents any-

thing from a dismal failure up to a near pass, whereas passing the

item involves barely passing up to passing with the greatest of ease"

Cp. 191). Thus, the biserial correlation is usually favored over the

point biserial correlation as a measure of item discrimination. Also,

the biserial is often chosen over the point biserial because the

magnitude of the point biserial correlation for an item is not in-

dependent of the item difficulty (Davis, 1951; Henrysson, 1971;

Swineford, 1956) . Specifically, values of the point biserial are

systematically depressed as p approaches the extremes of .00 or 1.00.

Lord and Novick (196S) have pointed out that because of this bias, the

point biserial correlation tends to favor medium difficulty items over

easy or very difficulty items.

The formulae for the biserial and point biserial correlation

respectively are (Magnusson, 1966, p. 200 § 203):

^bis =

17

One of tlie main objectives of classical test theory is to improve

the internal consistency of the test under construction where internal

consistency was defined as the extent to which all items are measuring

the same ability. To ensure high internal consistency the random error

in the test must be minimized. As stated previously in Equation 2,

reliability, in the classical model, was defined as:

r = "t^ = 1 - °e2XX

0^2 0^2

Thus, the relationship among the test items can be noted in the

coefficient alpha formulae for estimating internal consistency for a

sample (Magnusson, 1966, pp. 116-117):

2r = n , 1 - ESi , ,^.XX

_ ^- (5)

orT

r ..- " ^ik , (6)XX

S2

^X

2where n is the number of test items, Z S. is the sum of the item

1

2variances, S.," is the variance of the test, and C, is the mean of the

X ik

item covariances. By comparing equation 2 witli 5, it is seen that the

2sum of the unique item variances is used as an estimate of cr ", and that

when the unique item variation is minimized internal consistency will

be high. Furthermore, the mean of the item covariances (equation 6)

2serves as an estimate of a . 'Hie size of the covariance term is in

turn determined by the intercorrelations and standard deviations of

the items (Magnusson, 1966). Therefore, internal consistency is directly

dependent upon the correlation amons the items in the test.

18

The item discrimination index provides a measure of how well i'n

item contributes to what the test as a whole measures. W\en items with

tlie highest item-test correlations are selected, the homogeneity of

the test is increased; tliat is, o ~ is increased. So it is the item

discrimination that directly affects test reliability. V/lien items with

low item-test correlations are eliminated, the remaining item inter-

correlations are raised. Wlien item-test correlations are high, the test

is able to discriminate between higli and low scorers and hence internal

consistency is increased. If too few items are discarded in an item

analysis tlie internal consistency of tlie test tends to decrease because

items with little power of measuring what tlie entire test is intended

to measure will dilute the measuring power of the efficient items

(Beddell, 1950).

Research Related to Classical Item Ajialysis in Test Development

Several articles have been published concerning standards for item

selection to maximize test validity and increase internal consistency.

Flanagan (1959J stated two considerations in selecting test items:

(a) the item must be valid, that is, it should discriminate between

high and low scorers, and [b) the level of item difficulty should be

suitable for the examinee group. Gulliksen (1945) agreed with Flanagan

on these two points and added a third; items selected with p = .50 would

produce tlie most valid tests; liowever, Gulliksen noted that current

practice was opposed to selecting items with difficulty near .50. Test

developers were selecting items based upon spreading difficulty indices

over a broad range

.

Several studies have been conducted to examine the effects of

varying item difficulty on test development. Brogden (1946), in a

•

19

study of test homogeneity, has shown empirically that a test of 45

items witli varying levels of item difficulty produced a reliability

of .96 (measured by tlie Kuder-Richardson^ formula). However, a

similar but longer test of 153 items, that had item difficulties at

.50 for all items, produced reliability of .99. Thus, Brogden con-

cluded tliat effective item selection was based more on selecting a

test with fewer items that possessed varying difficulty, than a longer

test with equal item difficulty.

Davis (1951), in coiimientina on item difficulty, stated that if

all test items had a difficulty of .50 and were uncorrelated then

maximum discrimination was acliieved. But when test items were cor-

related, maximum discrimination would only be achieved when the

difficulty index for all test items was spread out, e.g., several

difficult items, several easy items, and several items with difficulty

near .50. Davis recommended the latter procedure for test development

because test items are usually correlated to some degree. Davis also

recognized the need for tlie approval of subject matter specialists in

addition to statistical criteria in item selection.

In a study of test validity, Webster (1956) foimd results similar

to Brogden (1946), but different from Gulliksen (1945). By selecting

fewer items with high discrimination indices and varying item difficulty

levels, a more valid test was produced. Webster's results indicated

that a test of 178 items with difficulty indices near .50 had a validity

coefficient of .66. However, a test of 124 similar items with varying

item difficulties had a validity coefficient of .76, statistically

significant at p < .03 (based on £ to £ transformations).

20

Myers (1962), concerned by the current practice of selecting

items based on varying item difficulties instead of the, theoretical

idea of p = .50, compared tlie effect of the current practice to the

theoretical idea on reliability and validity of a scholastic aptitude

test. Tlie ideal item difficulty ranged from .40 to .74 in what he

called the peaked test. Items selected by the current practice were

outside the above range, and Myers called this the U-shaped test.

Two sets of il -Tis were selected for the peaked test and the U-shaped

test, four tests in all. Myers reported no statistically significant

differences in test validity when the different tests were correlated

with freshman grades. Test reliability was statistically significant

at P < .02 (using the Wilcoxon matched pairs sign test) in favor of the

peaked test. The reliability of the peaked test was .69. The

reliability of the U-shaped test was .65. The author noted that the

results above were based on a 24 item test, and that when test length

was projected to 48 items (via Spearman- Brown Prophecy Formula) there

were no significant differences in test reliability. Tlie studies of

Brogden (1946) and IVebster (1956) indicate that selecting items of

varying item difficulty tends to increase internal consistency and test

validity. Tlie results from Myer's (1962) study indicated just the

opposite, that item difficulty near .50 produced the more internally

consistent test. But this was only true for a relatively short test

of 24 items, and that when the test length was projected to 48 items,

there were no differences in the reliability of cither test based upon

the two methods of selecting items.

21

Simplified Methods of Obtaining Item Discriminations

A second major group of articles on classical item analysis has

dealt witli simplified metliods of obtaining indices of item discrimination.

Because of the lack of computers in the early years of test develop-

ment many psychometricians concerned themselves with devising tables

to provide quick estimates of item discrimination. Kelley ("1939)

found that in the computation of item discrimination only 54 percent of

the examinee group (based on total test score) needed to be used.

Considering the top 27 percent and the bottom 27 percent of the test

scorers resulted in a considerable savings in computational time.

Flanagan (1939) developed a table of item discriminations to estimate

tlie PPM correlation between item and test score based on Kelley's

extreme score groups of top and bottom 27 percent.

Fan (1952) developed a table for the estimation of the tetrachoric

correlation coefficient using the upper and lower 27 percent of the

scorers, ilie tetrachoric correlation is similar to the biserial

correlation, where the correlation is between two variables, which are

assumed to have a normal and continuous underlying distribution, but

have been artifically dichotomized.

Guilford (1954) presented several short cut tabular and graphic

solutions for estimating various types of correlation coefficients to

measure test item validity. Tliese methods result in saving a consider-

above amount of time when one is forced to use hand calculations.

Today these short cut methods can be used by classroom teachers who often

do not have the aid of calculators or computers. However, many test

constructors still use these classical methods of item analysis even

22

though computers are available with which more sopliist icated item

analytic techniques such as factor analysis or latent trait models

can be used.

Item Analysis Procedures for the Factor Analytic Model

Charles Spearman [1904) proposed a theory of measurement based on

the idea that every test was composed of one general factor and a num-

ber of specific factors. In order to test liis idea Spearman developed

the statistical procedure known as factor analysis.

"Factor analysis is a method of analyzing a set ofobservations from their intercorrelations to determinewhether tlie variations represented can be accounted foradequately by a number of basic categories smaller thanthat with which the investigation started" (Fruchter,1954, p. 1).

Factor analysis is a mathematical procedure which produces a linear

representation of a variable in terms of other variables (Harman, 1967).

In the case of test items being factor analyzed, a matrix of item

intercorrelations is obtained first. Subsequently, the matrix of item

correlations is submitted to tlie factoring process. There are two

basic alternatives within the framework of factor analysis for analyzing

a set of data: coimnon factor analysis, based on the work of Spearman

and later Thurstone (1947); and principal components, developed by

Hotelling (1953). Tlie major distinction between the two methods relates

to the amount of variance analyzed, e.g., the values placed in tlie

diagonal of the intercorrelation matrix. Factoring of the correlation

matrix with unities in the diagonal leads to principal components, while

^ . 4factoring the correlation matrix with communalities in the diagonal

4,''

The communality (h~>, of a variable is defined as the sum of thesquared factor loadings h" = a. + a.^^ + ... a." (llarman, 1967, p. 17),see formula 8. J 3- jn

23

leads to common factor analysis (llarman, 1967) . If it is of interest

to know what the test items share in common, a common factor solution

is warranted. But if it is of interest to make comparisons to other

tests or other test development procedures, a principal components

solution is warranted. Since the present study was initiated to com-

pare three different test development techniques, a principal com-

ponents solution was used in this study to analyze the date under the

factor analytic model.

The linear model for the principal components procedure is

defined as (Harman, 1967, p. 15):

Z.. = a.,F, + a.-F- + . . .a. F . (7)ji jl 1 j2 2 jn n

Z.. is the variable for item) of interest, and a., is the coefficient,Jl jl

or more frequently referred to as the loading of variable Z.. on com-

ponent, (F,). An important feature of principal components is that the

extracted components account for the maximum amount of variance from

the original variables. Each principal component extracted is a linear

combination of the original variables and is uncorrelated with sub-

sequent components extracted. Thus, the sum of the variances of all

n principal components is equal to the sum of the variances of the

original variables (Harman, 1967). According to Guertin and Bailey

(1970), the principal components solution was designed basically for

prediction, hence the need to use the maximum amount of variance in a

set of variables

.

Since factor analysis is based upon a matrix of intercorrelations

,

it is important tliat care be taken in selecting the appropriate

coefficient. Several item coefficients are available: phi, phi/phi

max, and the tetrachoric correlation coefficient. Carroll (1961)

24

pointed out several problems concerning the clioice of a correlation

coefficient to be used in factor analysis. The phi coefficient (used

where both variables are true dichotomies) was found to be affected

by disparate marginal distributions and often imderestimated tlie PPM.

The phi/phi max coefficient was developed to correct for the under-

estimation of phi, but the correction is not enoush to counter the

effect of extreme dichotomizations. Carroll recoiranended the tetrachoric

coefficient as being the least biased by extreme marginal splits

providing the variable uiider consideration was nori:ially distributed in

the population. Wherry and Winer (1955) had made conclusions similar

to Carroll, but went on to say that when the normality assumption was

met and the regression of test score on the item was linear the PPM

and tetraclioric are identical. Tlie tetrachoric correlation was used in

the present study to obtain item intercorrelations

.

Research Related to Factor Analysis in Test Development

The early use of factor analysis to construct and refine tests

was suggested by the work of McNemar (1942) in revising the Stanford-

Binet scales, and Burt and John (1943) in analyzing the Terman-Binet

scales.

Several contemporary psychometricians have advocated the use of

factor analysis in developing unidimensional tests (Cattell, 1957;

Hambleton and Traub, 1975; Henrysson, 19b2; Lord and Novick, 1968). A

unidimensional test was defined briefly in the introduction to this

chapter, but a more precise definition is warranted. Lumsden (1961)

noted that a unidimensional test can be determined by the examinee

response patterns. If the test items are arranged from easiest to

hardest, person^ who misses item will miss all the other items, and

25

person-, who gets item correct but misses item^ will miss all the

subsequent items and so on. The above statement assumes infallible

items. However, most tests constructed today contain fallible items,

thus the response pattern will be disturbed by random error. Lumsden

suggested in developing unidimensional tests factorial ly that the items

be carefully selected on empirical grounds, tlms reducing the problem of

too many heterogeneous items and the possibility of obtaining multiple

factors. By preselecting items one increases the chances of the items

converging on one factor.

The importance of developing unidimensional tests is demon-

strated most clearly in considering the concepts of test reliability

and validity. For a test to be valid it must actually measure the trait

it was intended to measure. For a test to be reliable it must provide

similar results upon repeated measurement. It should be easier to

estimate these two important aspects of a test when the test is

unidimensional than when the test is multidimensional, hence the use of

a unidimensional test in the present study.

Cattell (1957) has suggested that in the development of a factor

homogeneous scale, one should preselect items, carry out a preliminary

factor analysis, then select for further analysis those items which

load on the first factor. Cattell defined an index of unidimensionality

as the ratio of the variance of the first factor to the total test

variance. This index has no set criterion and the sampling distribution

is unknown.

Comparison of Factor Analysis to Classical Item Analysis

One measure of item validity, the biserial correlation was described

for classical item analysis procedures. This same index is also obtained

26

by factor analysis. U'lien tiie test items are factor analyzed, the factor

loading a... is the item- factor association that is considered ao ij

measure of item validity, e.g., the liigher tlie factor loading, the

greater the relationship between the item and the factor it iieasures.

Tlie factor loadings can be viewed as similar to the biserial correlations

discussed under classical test theory. Tliis relationship between

factor loadings and biserial correl: tions has been discussed by several

authorities (Guertin and Bailey, 197;'; Henrysson, 1962; Richardson,

1936).

Factor analysis as an item analytic technique was not realistically

possible for most psychometricians until the advent of high speed

computers. Guertin and Bailey (1970) iiave predicted that with the

increasing use of computers factor analysis will replace classical item

analysis as a test development technique. Because it is possible for

a test to reach the highest degree of homogeneity and yet be factorial ly

a very odd mixture of factors (Cattell and Tsujioka, 1964), classical

item analysis alone is not sufficient to determine if a test is

unidimensional . However, factor analysis not only provides a measure

of item-test correlation (the factor loading), it also provides an

indication of how many items form a unifactor test. Thus, factor

analysis has been advocated as a superior technique to classical item

analysis (Guertin and Bailey, 1970). Using factor analysis in test

development, psycliometricians liave advanced beyond an independent

analysis of item intercorrelations to a simultaneous analysis of item

intercorrelations with other individual items to obtain a measure of

test unidimensionality and item-factor association.

27

However, there is an inherent flaw in factor analysis as there

was in classical item analysis in test development. Tlie flaw is that

both procedures are sample dependent. Wien an item analysis procedure,

or any procedure in general is sample dependent, it means that the

results will vary from group to group, fflien the groups are very

dissimilar, there is much variability. Gulliksen (1950) noted that a

significant advance in item analysis tlieory would be made when a

method of obtaining invariant item parameters could be discovered. To

that end latent trait theory is an attempt to identify invariant

item parameters.

Item Ajialvsis Procedures for the Latent Trait Model

Latent trait theory specifies a relationship between the observable

examinee test performance and the unobservable traits or abilities

assumed to underlie performance on a test (liambleton et al., 1977). The

relationship is described by a mathematical function; hence latent

trait models are mathematical models. As noted earlier, there are four

major latent trait models for use with dichotomously scored data: the

normal ogive, and tlie one-, two-, and three-parameter logistic models

(Hambleton and Cook, 1977; Lord and Novick, 196S) . All four models are

based on the assumption that the items in the test are measuring one

common ability and that the assumption of local independence exists

between the items and examinees. These two assumptions imply that a

test which measures only one trait or ability will have less measurement

error in tlie test score than a test that is multidimensional, and that

tl;e response of an examinee to one item is not related to his response

on any other item. Ifliere the latent trait models begin to differ is

with respect to the shape of their item characteristic curves.

28

Tlxe normal ogive, develoj^ed by Lord (1952a, 1955a), produces

an item characteristic curve based on the following formula:

a (e - b ) (8)Pg CQ) = / f ^ e(t)dt,

where P^ (6) is the probability that an examinee with ability 6 correctly

answers item g, 6(t) is the normal density function, b represents item

difficulty and a represents item discrimination.

The item characteristic curve of thertwo-parameter logistic model

developed by Birnbaum (1968) has the same shape as the normal ogive,

and Baker (1961) has shown them to be equivalent mathematical procedures.

Tlie shape of the item characteristic cui-ve of the two-parameter

logistic function is developed from the following formula:

Dag(e - b )

'^ ''' -, . e

^-^ '^ - V '"'

P (9), ^ and b have the same interpretation as in the normal ogive.& o t>

D is a scaling factor equal to 1.7 (the adjustment between the logistic

function and normal density functioii) , and e is the natural log function.

In Figure la the shape of the normal ogive and the two-parameter

logistic curve has been illustrated. In the Figure, item A is more

discriminating tlian item B as noted by the steepness of the slopes.

Ilie. three-parameter logistic model also developed by Birnbaum (1968)

includes as an additional parameter, an index for guessing. The

mathematical form of the three-parameter logistic curve is denoted,

^.Dag(9 - b )

S ^"^'8 ^* ^g'

1 + e^-^^ ^^ - ^^

Tlie parameter c the lower asympotote of the item characteristic curve,

represents the probability of low ability examinees correctly answering

29

-T— r^ r-3.0 -2.0 -1.0

ABILITY CONTINUUM

NORMAL OGIVE & TWO-PARAMETER LOGISTIC CURVE (a)

1.00 -T-

T-—r——r r-1.0 1.0 ZO

ABILITY CONTINUUM

13.0

THREE-PARAMETER LOGISTIC CURVE (b)

uj 1.00 T< CO

o2 .75

> COr un °= .50 HCD •-

o '^.25

T-3.0 -ZO -i.O 1.0

ABILITY CONTINUUM

RASCH ONE- PARAMETER LOGISTIC CURVE (c)

FIGURE L HYPOTHETICAL ITEM CHARACTERISTIC

CURVES FOR THE FOUR LATENT TRAIT

MODELS.

30

an item (llambleton et al., 1977). In Figure lb the shape of the

three-parameter logistic curve has been illustrated. In tlie Figure,

item A is more discriiainating and has less guessing involved than item B.

The one-parameter logistic model, developed by Rascli (1960)

is commonly referred to as the Rasch model. The Rasch model, though

similar to the other latent trait models, was developed independently

from the other models. The Rasch model is based upon two propositions:

(a) the smarter an examinee, the more likely he is to answer the item

correctly, and (b) an examinee is more likely to answer an easy item

correctly than a difficult item. Matliematical ly the above propositions

can be stated in terms of odds or probability of success on an item.

Tlie odds of an examinee with ability t) correctly answering an item with

difficulty <', is given by the ratio of G to c, (Rasch, 1960):

odds = 8 . (11)

The derivation of equation 11 was presented in Appendix A. Equation 11

re formally written in the following equation is the Rasch model.mo

ef\ - *i^ (l^)

Inequation 12, tlie probability of examinee k making a correct response

to item i, noted X = 1 , given an examinee of ability 3, (where 3, is thek k

log transformation of 0) taking an item of difficulty 6. (where 6. is the1 1

log transformation of cj is a function of the difference between the

examinee's ability and the item's difficulty. The derivation of

equation 11 to equation 12 is presented in Appendix A.

The assumptions for the Rasch model were discussed in Chapter I.

Essentially the three assumptions are as follows:

31

1. There is only one trait underlying test performance.

2. Item responses of each examinee are statistically independent.

3. Item discriminations are equal.

The first two assumptions can be checked by conducting a factor analysis

of the test items as suggested by Lord and Novick (1968) , and Hambleton

and Traub (1973) . Tlie assumptions are met if one dominant factor

emerges from tlie analysis. Tlie tliird assumption can be checked by

plotting item characteristic curves for each item. In Figure Ic the

item characteristic curves for two hypothetical items based on the Rasch

model have been illustrated. The difficulty for items A and B is .Sand

L5 respectively (point where p = .50), and the discriminations of the

two items are equal. The assumption that all items have equal dis-

criminations is quite restrictive; however, Rentz (1976) demonstrated,

in a simulation study, that the item slopes can deviate from 1 (where

all slopes are equal) + .25 and still fit the model. In a similar

simulation study, Dinero and Haertel (1976) concluded that the lack of

an item discrimination parameter in the Rasch model does not result

in poor item calibrations when discriminations are varied as much as .25.

The estimates for the Rasch parameters 3, and 5., examinee abilityk 1

estimate and item difficulty estimate respectively, are sufficient,

consistent, efficient, and unbiased (Anderson, 1973; Bock and Wood, 1971)

That is, the examinee's test score will contain all the information

necessary to measure the person ability parameter S, , and the sum of the

right answers to a given item will contain all the information used to

calibrate the item parameter (S . (Wright, 1977). Of tlie latent trait

models, the Rasch model is imique in tliis respect.

Tlie matliematical rationale of the Rasch noJcl is based upon the

separation of the ability and item difficulty parameters. As shown

in Appendix A, the estimation of- the item parameters is independent of

the distribution of ability and ability independent of the distribution

of item difficulty (Rasch, 1966). Several studies have demonstrated

this (Anderson et al., 1968; Tinsley and Dawis, 1975; Uliitely and

Dawis, 1974; Miitely and Dawis, 1976; IVright, 1968; Wright and

Panchapakeson, 1969). The separation of the ability and item param-

eters leads to what Rasch has termed specific objectivity. Specific

objectivity relates to the fact that the measurement of a person's

ability is not dependent upon the sample of items used, nor the

examinee group in which a person is tested. Once a set of items has been

calibrated to the Rasch model, any subset of tlie calibrated items will

produce the same estimate of the examinee's ability. This type of

objectivity is possessed by the physical sciences and the goal toward

which mental measurement should be aimed in the future. Toward the

goal of objective measurement several researchers have conducted

empirical studies comparing classical factor analytic test development

procedures to the latent trait models, and also comparisons have been

made between the various latent trait models.

Research Related to Latent Trait Models in Test Development

Baker (1961) conducted one of the earlier comparative studies

between two latent trait models. He compared the effect of fitting the

normal ogive and the two-parameter logistic model to the same set of

data, a scholastic aptitude test. The two-parameter model as well as

the noi-mal ogive provide item difficulty and item discrimination estimates,

33

The empirical results suggest there is little difference between the

two procedures as measured by a chi-square test of fit. However, Baker

noted the computer running time of the logistic model was one-third

that of the ogive model, thus he concluded the logistic model was

more efficient in terms of cost than the ogive.

Hambleton and Traub (1971) compared the efficiency of ability

estimates provided by the Rasch model and the two-parameter model to

the three-parameter logistic model using Bimbaum's concept of infor-

mation (1968). The three-parameter model provides item difficulty

and discrimination estimates as well as accounting for guessing on

each item. Eleven simulated tests of fifteen items each were generated

varying item discrimination and degree of guessing. ITie authors

sought to determine how efficient the one- and two-parameter logistic

models were under these conditions taking the three-parameter model to

be the true model. Tlie results indicated that wlien guessing was a

factor the three-parameter model was most efficient in providing ability

estimates, but when guessing was not a factor all models were equally

efficient. Since the Rasch model has fewer parameters to estimate, hence

it takes less computer time to run than the other two models, it would

be preferred in the absence of guessing. In considering item

discrimination, when the guessing parameter was set to zero, the Rasch

model was as efficient as the two-parameter model when item discrimination

varied from .59 to .79. As item discrimination deviated from this range

the two-parameter model was more efficient.

Hambleton and Traub (1975) compared the one- and two-parameter

models with three sets of real data (the verbal and mathematics subtests

of a scholastic aptitude test used in Ontario (items = 45 and 20

34

respectively), and tlie veii)al section of the Scliolastic Aptitude Test

(SAT, items - 80). llieir results indicated that generally tlie two-

parameter model fit the data better than the one-parameter model. The

loss in predicting performance was greatest on tlie shorter mathematics

test and smallest on tlie longer SAT. iliese findings confirm Birnbaum's

conjecture (1908, p. 492) that if the number of items in a test is very

large tlie inferences that can be made about an examinee's ability will

be much the same whether the Rasch model or the two-parameter logistic

model is used. The authors questioned whetlier the gain obtained with the

two-parameter model is worth tlie increased computer cost of estimating

the item discrimination parameter. Based on the results of these studies,

it is concluded that the Rascli model is tlie most efficient of the latent

trait models and hence will be used in comparison to the more traditional

methods of test development included in the present study.

Comparison of tlie Rasch Model to Factor Analysis

Two recent studies have been completed comparing the Rasch model to

factor analysis. Anderson (1976) posed two questions concerning the

Rasch model and factor analysis: (a) what types of items would be

excluded in terms of difficulty and discrimination using Rasch and

factor analysis as item analytic teclmiques, and (b) what effect would

the two procedures have on validity? Anderson chose to use 235 middle

school students' responses to a 15 item Likert-type scale that was

dichotomized for use witli tlie Rasch model and the factor analytic

procedures. A principal component factor analysis based upon tetrachoric

correlation coefficients w a s compared to the Rasch model using the

CALl-'lT computer program (Wright and Mead, 1975). Only items fitting

the model were used. His results indicated that the Rasch procedure

35

eliminated the more difficult items and the factor analytic procedure

eliminated the easier items; a statistically significant difference as

determined by clii-square test at £ < .01. For item discriminations the

Rasch procedure eliminated very low and very high item discriminations,

while the factor analytic procedure tended to reject only very low

discriminations. Tlie difference here was not statistically significant.

The second question of test validity showed very similar results for

the two procedures wlien test score was correlated with course grade

point average.

In a similar study Mandeville and Smarr (1976) developed a two

stage design. First they compared the Rasch procedure to factor analysis,

then they combined the two analytic procedures. The authors felt the

combined approacli would be a more effective item analytic approach

than any single method in detei^mining whicli items fit the Rasch model.

Two cognitive data sets (one standardized and one classroom) and one

simulated set were used in the study. A rotated principal axis factor

analysis based upon phi correlation coefficients were compared to the

Rasch model using tiie CALFIT program.

The results indicated that for the standardized and simulated data

sets the double procedure of factor analyzing the items, then submitting

only the items loading on the first factor to the Rasch procedure was

not really useful. The Rasch procedure alone was just as effective as

the double procedure in selecting items that fit tlie model.

For the classroom data set the investigators found that 92 percent

of the items fit the Rasch model, but upon factor analyzing these

items only seven percent of the total test variance was associated with

tl)e first factor. Their results tend to indicate that factor analysis

56

and the Rascl; procedure do not always identify the same unidimensional

trait underlying test performance. However, the results of the

Mandeville and Smarr study may be' suspect for three reasons. First,

the plii coefficient, which can be seriously affected when p and q

take on extreme values, was used as a basis to form the intercorrelation

matrix that was factor analyzed. The greater the difference in p and q

the smaller will be the maximum correlation, hence very easy and very

difficult itej-.is will have systematically lower coefficients and will

tend to bias the results of the analysis in favor of moderately difficult

items. Second, the factor analysis was based on a principal axis

solution, using some value less than 1.00 in tlie diagonal hence less

variance is being used in the total solution for comparison with the

Rasch procedure that is utilizing all the test variance available.

Third, the principal axis solution was rotated so that the total

variance associated with the first factor has been distributed out

among the other factors and was no longer as strong as it once had been.

Suimnary

In the development of tests based upon classical item analysis

two main statistics are used in reviewing and revising test items, e.g.,

item difficulty and item discrimination. The item discrimination index

provides information as to the validity of the item in relation to total

test score, while item difficulty indicates ]:ow appropriate the item was

for the group tested. A serious limitation of classical item analysis

is that the statistics obtained for examinees and items are sample

dependent (Hambleton and Cook, 1977; bright, 196S)

.

The same problem of sample dependency also exists for factor

analysis. However, factor analysis is viewed as a superior technique

37

to classical item analysis for two reasons: (a) factor analysis

compares item intercorrelations with other items simultaneously,

and (b) factor analysis provides an indication of how many factors

or abilities the test is measuring. Also in factor analysis, the

factor loading is comparable to the item discrimin ition index of

classical item analysis, thus providing a measure of item validity

for eacli item on each factor in the test.

Not until the development of latent trait models was a solution

suggested to the problem of sample dependency of tiie statistics for

items and examinees. The Rasch model in particular has been shown to

provide item statistics that are independent of the group on which they

were obtained, as well as examinee statistics that are independent of

the group of items on wliicli they were tested. Tliis feature of the

Rasch model provides for more objective mental measurement.

The Rasch model lias been compared to other latent trait models

and has been shown to be as efficient in many cases as the more complex

models. Tiie Rasch model has also been compared with factor analytic

procedures in determining test unidimensionality , validity, and types

of items retained and excluded by tlie two procedures. Missing from

this review is a comparative study of the three item analytic techniques

using the same data base and a comparison of the efficiency of tests

developed from the three techniques across ability levels. Also missing

from the literature is the effect of varying sample size and number of

items as well as the kinds of items each of the three procedures would

eitlier retain or exclude in test development.

38

It is apparent that an empirical investigation into these areas

seems warranted to determine which procedure under tlie various

conditions would produce the superior test in terms of internal

consistency and efficiency. It was for this reason that the present

study was undertaken comparing the three methods of classical item

analysis, factor analysis, and the Rasch model used in test development,

The design of tlie study is described in Chapter III.

CHAPTER III

METHOD

An empirical study was designed to compare the effects of three

methods of item analysis on test development for different sample sizes.

The three methods of item analysis studied were classical item analysis,

factor analysis, and Rasch analysis. The sample sizes used to compare

the tliree item analytic methods were 250, 500, and 995 subjects. The

study was designed in three phases: (a) item selection, (b) a double

cross-validation of the selected items, and (c) statistical analyses

of the selected items. For each item analytic procedure two tests

were developed, a 15 item test, and a 30 item test. Four dependent

variables were obtained for each test: (a) an estimate of internal

consistency, (b) the standard error of measurement, (c) item difficulty,

and (d) item discrimination. A description of the subjects, instrument

used, research design, and statistical analyses is presented in this

chapter.

The Sample

In the fall of 1975, all high school seniors in the State of

Florida (N = 78,751) were tested as part of the State assessment program.

The population was from 435 high schools throughout the state. From

this population a 1 in 15 systematic sample of 5,250 subjects was

chosen (Mendenhall , Ott, and Scheaffer, 1971). A systematic sample was

selected to ensure samples from every high school in the state. The

39

40

types of data obtained on each suoject were sex, race, item responses,

and total score.

The data file was edited to remove those subjects who cither

answered all the items correctly or incorrectly. The rationale for

this procedure was that the Rasch model cannot calibrate items when

a person has a perfect score or the alternative, when a person has no

items correct [Wright, 1977). Through the editing procedure 15 subjects

were removed, thus the available sajnple size was 5,255. Because such

a small number of subjects were removed, it seems unlikely that the

elimination of these subjects would bias the results in favor of any

of the three item analytic techniques.

The Instrument

Tlie instrument selected for use in this study was the Verbal

Aptitude subtest of the Florida Twelfth Grade Test, developed by the

Educational Testing Service. It is a statewide assessment battery

which has been administered every year since 1935 (Benson, 1975). The

Verbal Aptitude subtest is comprised of 50 verbal analogies, in a

multiple choice format, from which a single score based on the number

of items correct is reported. Descriptive information on the Verbal

Aptitude subtest for the population tested in 1975 is presented in

Table 1.

This particular instrument was selected for three reasons. First,

it is a cognitive measure of verbal ability and much of classical

test theory has been build upon tests in the cognitive domain. Second,

it is similar to and hence representative of other national aotitude

tests used for college admissions. Tliird, it has a large data pool

from which to sample.

41

TABLE 1

DliSCRIPTIVH DATA ON THE VERBAL APTITUDE SUBTESTOF THE FLORIDA TWELFTH GRADE TEST

1975 ADMINISTRATION

Number of Schools = 435 Number of Students - 78,751

Number of items 50Mean 25.95Standard Deviation 8.23Reliability^ .88

Standard Error of Measurement 2.85

Note : Data obtained from the Florida Twelfth Grade TestingProgram, Report No. 1-75, Fall 1975,

Reliability based on the split-half method, and corrected bythe Spearman-Brown fox'mula.

42

Classical test theor)' lias been built mainly around the development

of cognitive tests. Therefore, it seemed desirable to compare the

new procedures of latent trait theory, via the Rascli model to the

procedures of classical test tlieory, e.g., factor analysis and classical

item analysis by using a cognitive test. Thus, the results may be

more generalizable to the major type of tests developed by practitioners

in the field.

The Procedure

Design

The sample of 5,235 v.'as divided into nine systematic samples in

the following manner:

Group = three independent samples of 250 students each;

Group = three independent samples of 500 students each;

Group_ = three independent samples of 995 students each.

From the initial editing of the data file, previously described, 15

subjects were removed from the total sample of 5,250. Therefore,

it was decided that this loss of subjects would only affect Group

since it was the largest. Thus, the number of subjects in each of

the three independent samples was reduced by five, resulting in three

independent samples of 995 subjects each.

The purpose of obtaining the three separate samples for t]>e three

groups was to insure that each item analytic and double cross-validation

procedure used an independent sample, so that tests of statistical

significance could be perform.ed. The scheme shown in Table 2 was used

to obtain tlie nine samples. In the present study the independent

variables were sample size and item analytic procedure.

43

TABLE 2

SYSTEMATIC SAMPLING DESIGN OF THE STUDY"

N = 5,235

44

The item data were analyzed in thi-ee pliases : (a) selection of the

items, (b) computation of item and test statistics for selected items

on double cross-validation samples, and (c) statistical analyses of

item characteristics to test tlie hypotheses.

Item Selection

The three independent samples, within each of the groups of subjects

(N = 250, N = 500, N = 995), were submitted to one of the three item

analytic procedures (in accordance with Table 2) in order to select a

specified number of items, e.g., the "best" 15 and 30 items. Each of

these two sets of items comprised two separate tests; however, all of

the items on the 15 item tests were always included on each of tlie 30 item

tests. A different process for selecting the items was used with each

item analytic technique, and has been described in the following three

sections.

Classical item analysis . The definition of the "best" items was

based on the numerical magnitude of the items' biserial correlations.

The biserial correlation was defined as the correlation between the

artifically dichotomized item score (1 or 0) and total test score.

In using the biserial correlation the assumption was made that the

artifically dichotomized variable (the item) had a continuous and normal

distribution (Magnusson, 1966)

.

In order to obtain biserial correlations for the items under the

classical item analysis procedure, the 50 vei'bal items were submitted

5to the item analysis program, GITAP for each of the three sample sizes.

The Generalized Item Analysis Program (GITAP) is a part of thetest analysis package developed by F. B. Baker and T. J. Martin,Occasional Paper No. 10, Michigan State University, 1970.

45

The 15 and 30 items with the highest biserial correlations were selected

as the best items from the total subtest. Item difficulties were also

obtained for the "best" 15 and 30 items selected. Item difficulty has

been defined as the proportion of persons getting a particular item

correct out of the total number of persons attempting that item

(Mehrens and Lehman, 1973).

Factor analysis . Item selection based on factor analysis was

accomplished using the computer programs developed for the Education

Evaluation Laboratory at the University of Florida. Tliese programs

have been described by Guertin and Bailey (1970). The present study

was concerned only with the items that load on the first principal

component, in order to adhere to the unidimensionality assumption of

the test. The principal components analysis was based on a matrix of

tetrachoric item intercorrelations with unities in the diagonal.

Tlie tetrachoric correlation was chosen to produce the intercorrela-

tion matrix for the same reason tlie biserial correlation was chosen:

Knowledge of an item was assiomed to be normal and continuously dis-

tributed. In the case of the tetrachoric correlation each item (scored

1 or 0) was correlated with every other item.

The 15 and 30 items with the highest loadings on the first unrotated

principal component were selected from the total subtest. These com-

ponent loadings are analogous to biserial correlations previously

described, where the loading refers to the relationship of the item to

the principal component or factor (Guertin and Bailey, 1970, Henrysson,

1962).

Rasch analysis, llie selection of items based on the Rasch model

was accomplished in two stages. First, in order to check the assumption

46

)f a unidimensional test, a factor analysis using a principal components

solution was used. Items were selected with loadings between .39 and

79 on the first unrotated factor-, to hold the discrimination index

of the items constant. Hambleton and Traub (1971) have shown that the

efficiency of a test developed using the Rasch model will remain very

high (over 95 percent) when the range on the discrimination index was

held between .59 and .79. Second, the items selected from the principal

components solution using the above criteria were submitted to a

Rasch analysis using the IbICAL i/rogram (Wright and Mead, 1976) . Items

were selected based upon the mean square fit of the items to the

Rasch model. The best 15 and 30 items fitting the model were chosen

from the total subtest, and their corresponding item difficulties

reported.

Double Cross-Validation

A double cross-validation design (Mosier, 1951) was used to obtain

item parameter estimates for the best 15 and 50 items selected by the

three item analytic techniques for the three sample sizes. In this

study a 5 X 3 latin square was used to reassign samples. This procedure

ensured that the estimates of the item parameters would be based

upon a different sample of subjects than the original sample used to

identify the best items. Each item analytic technique was randomly

reassigned, using a latin square procedure (Cochran and Cox, 1957,

p. 121), to a different sample within each of the three groups (N - 250,

N = 500, N = 995) . The double cross-validation design is shown in

Table 5.

The best 15 and 30 items selected by each item analytic procedure

in the first phase of the study, were submitted to a standard item

47

TABLE 3

DOUBLE CROSS-VALIDATIOM DESIGN OF TliE STUDY

SampleGroup Number^

NumberSelected

Item AnalyticProcedure

Double Cross-Validation Procedure

GrouD1

Group

Group.

1

2

3

4

5

6

7

8

9

250 Classical250 Factor Analysis250 Rasch

500 Classical500 Factor Analysis500 Rasch

995 Classical995 Factor Analysis995 Rasch

Factor AnalysisRaschClassical

RaschClassicalFactor Analysis

RaschClassicalFactor Analysis

The sample number is the same as referred to in Table 2.

Assignment to sample was based on a randomized 3X3 latinsquare procedure.

48

analysis program (GITAP) from wliich were obtained the dependent

variables in the study:

•indices of internal consistency as measured by the analysis

of variance procedure (Hoyt, 1941)

•the standard error of measurement

•item difficulty

•biserial correlations

By submitting the best 15 and 50 items selected by each item analytic

procedure in the study to a common item analysis program comparable

measures of the dependent variables were obtained.

Statistical Analyses

The third phase of the study focused on obtaining measures of

statistical significance for three of the dependent variables: internal

consistency, item difficulties, and biserial correlations. Only visual

comparisons were made for the remaining dependent variable, the standard

error of measurement. The internal consistency estimates from each test

were compared to the projected population value for tests of similar

length via confidence intervals as suggested by Feldt (1965). (Projected

population values were obtained using the Spearman-Brown Prophecy Formula.)

Item difficulties for the 15 and 50 best items were submitted to a

two-way analysis of variance, the two factors being sample size and item

analytic technique. This procedure was used to test for differences in

the types of items selected, in terms of item difficulty, by each technique.

If statistical significance was observed, with a = .05, Tukey's HSD

(honestly significant difference) post hoc procedure (Kirk, 1968) was

The analysis of variance procedure is appropriate only if thedistribution of the item difficulties and (transformed biserial correlations

49

employed to determine which item analytic teclinique(s) resulted in a

test with the highest item difficulties.

The biserial correlations were transformed to an interval scale

of measurement using a linear function of £ suggested by Davis (1946).

The linear transformation was based upon converting the biserial

correlation to z values, and then eliminating tlie decimals and negative

values of z by multiplying the constant 60.241 to each z_ value (Davis,

1946, pp. 12-15). Thus, the range of the transformed biserials ranged

between and 100. A two-way analysis of variance (sample size by item

analytic technique) was performed on the transformed biserial correlations

for the best 15 items. Tliis type of analysis was used to test for

differences in the types of items selected, in terms of biserial correla-

tions by each technique. If statistical significance was observed,

a = .05, Tukey's HSD post hoc procedure was employed to determine which

item analytic technique(s) resulted in higher transformed biserial

correlations.

The two-way analysis of variance and post hoc analysis, where

indicated, for the transformed biserial correlations was performed on

the 30 best items.

In addition to tests of statistical significance, a measure of

the efficiency of the 30 best items selected by each procedure was

compared for the sample of 995 subjects. Birnbaum (1968) defined the

relative efficiency of two testing procedures as the ratio of their

approximates normality and the variances are homogeneous (Ware and Benson,

1975) .

The analysis of variance procedure is appropriate only if the

distribution of the item difficulties and (transformed) biserial correla-

tions approximates normality and the variances are homogeneous (Ware and

Benson, 1975).

50

information curves. Lord (1974aJ lias described a procedure to compare

the relative efficiency of one test with another at different ability

levels. If two tests to be compared vary in difficulty, then the

relative efficiency of each will usually be different at different

ability levels (Lord, lC74b; 1977). In classical test theoiy it is

common to compare two tests that measure the same ability in terms of

their reliability coefficients, but this only gives a single overall

comparison, llie foiT.iula developed by Lord for relative efficiency

provides a more precise way of comparing two tests that measure the

same ability. The formula for approximating relative efficiency is

(Lord, 1974b, p. 248):

2

n c r ^ " X (n -x)f ., „.R.E. (y,x3 = y • X -x , (loj

n y (n -y)^2X ' ' y 'fy

where R.E. denotes the relative efficiency of y compared to x, n and n

denote the number of items in the two tests, x and y are tlie number-

2 2right scores having the same percentile rank, and f _ and f are the

squared observed frequencies of x and y. Lord lias suggested that

formula 13 only be used with a large sample of examinees and tests that

are not extremely short, hence this comparison was restricted to the

case where N = 995 and the 30 item test.

Tliree relative efficiency comparisons using the 30 item tests were

made: (a) the test based on factor analysis was compared to the test

based on classical item analysis, (b) the test based on the Rasch

analysis was compared to the test based on classical item analysis, and

(c) the test based on the Rasch analysis was compared to the test based

on factor analysis.

5]

Summary

An empirical study was designed to compare the effects of classical

item analysis, factor analysis, and the Rascli model on test development.

Item response data were obtained from a sample of 5,255 high school

seniors on a cognitive test of verbal aptitude.

The subjects were divided into 9 samples: three independent

groups of 250 subjects eacli, three independent groups of 500 subjects

each, and tlu-ee independent groups of 995 subjects each. The independent

groups were obtained so that tests of statistical significance could

be perfoi'ined.

The item response data were then analyzed in three phases. First,

the "best" 15 and 30 items were selected using each item analytic

technique. Under classical item analysis, the best 15 and 30 items

were selected based on the liigliest biserial correlations. For factor

analysis, the best 15 and 30 items were selected based on the highest

item loadings on the first (unrotated) principal component. The

selections of the best 15 and 30 items using the Rascli model were

based upon the mean square fit of the items to tlie model. These

procedures were used for each group of subjects. Second, a double

cross-validation design was employed to obtain estimates on the item

parameters for tlie best 15 and 30 items. The three item analytic

techniques were reassigned randomly to different samples of subjects

within each level of sample size. Then, the best 15 and 30 items

chosen by each method were submitted to a conmion item analytic procedure

in order to obtain estimates for comparing the three item analytic

methods. Third, a two-way analysis of variance and a Tukey post hoc

comparison test, when indicated, were used to test for differences in

52

the properties of items selected by each item analytic procedure. Also

confidence intervals were calculated to compare the internal consistency

estimates to a population value. In addition, tlie relative efficiencies

of the 30 item tests developed by eacl; item anal)'tic technique were

compared for the sample of 995 subjects.

••

CHAPTER IV

RESULTS

The study was designed to compare empirically the precision and

efficiency of tests developed using three item analytic techniques:

classical item analysis, factor analysis, and the Rasch model. The

following five hypotheses were generated to compare the three techniques:

1. There are no significant differences in the internal consistency

estimates of the tests produced by the three methods as the number of

items decreases when compared to the projected internal consistency

estimates for the population for tests of similar length.

2. There are no differences in the internal consistency estimates

of the tests produced by the three methods when the nunber of examinees

is decreased.

g3. There are no meaningful differences in the magnitude of

the standard error of measurement of the tests produced by the three

methods

.

4. There are no significant differences in the difficulties or

discriminations of the items selected by the three methods.

A meaningful difference was previously defined to be >_ 1.00.

53

54

5. Tliere are no differences across ability levels in the efficiency

of the tests produced by the tliree metliods.

The Verbal Ajjtitude subtest of the 1-lorida iwelfth Grade Test, was

used to test the hypotlieses. A sample of 5,235 examinees was

systematically selected from a population of 78,751. A demographic

breakdown of the sample by etl;nic origin and sex is presented in Table 4.

The data were analyzed and reported in the following manner: item

selection, double cross-validation, comparison of the 15 item tests on

precision and comparison of the 30 item tests on precision and efficiency.

These results were then summarized witli respect to tlie five h)-potheses.

Item Selection

The 50 items on the Verbal Aptitude subtest were submitted to each

of the tliree item analytic techniques. The means, medians, and standard

deviations of t'ne bi serial correlations and item difficulties, based

on classical item analysis, are presented in Table 5. These descriptive

statistics appear ecjuivalent across the varying sample sizes.

From tlie factor analysis, the percentage of total test variance

accounted for by the 50 verbal items on the first unrotated principal

component has been reported in Table 6. The percentage of variance

accounted for by the first principal component was obtained by summing

the squared item loadings and dividing by the total number of items.

The percentages of variance accounted for by the first principal component

in eacli sample were very similar. A check on the unidimensionality

of the test was made by rotating the principal components solution for

the sample of 995 subjects. Upon rotation, the results indicated

one dominate factor remained.

55

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Three statistics ai-e reported in Table 7 for the Rasch item

analysis procedure. For each sample size, the percentage of total

variance accounted for by the first unrotated principal conponent

and the means and standard deviations of the mean square fit statistic

and Rasch difficulties are presented for the items selected.

In order to select the best 15 and 30 items from the Rasch analysis,

all 50 items were submitted to a principal components solution. Tliis

procedure was used to ensure that the items selected measured one

trait, as required by the assimiption of test unidimensionality. As

noted in Table 1 , the percentage of total test variance accounted for

by the first principal component, based on 50 items, was nearly equal

for each sample size.

From the principal components solution only items with loadings

between .39 and .79 were selected for the Rasch analysis as suggested

by Hambleton and i'raub (1971), to adhere to the assumption of equal

item discriminations. Using this procedure the number of items (out

of 50) retained for the Rascli analysis varied slightly with sample

size; when N = 250, 53 items were retained, when N - 500, 35 items

were retained, and when N = 995, 33 items were retained. Tliese items,

loading between .59 and .79, were then submitted to the Rasch analysis

to obtain mean square fit statistics and Rasch item difficulties.

These st;;tistics have been reported in Table 7.

Wriglit and Panchapakesan (1969) developed a measure to assess the

fit of the item to the Rasch model. Tlie measure, defined as the mean

square fit statistic, is:

61

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2The quanity, x > defined above has approximately the chi square

distribution with degrees of freedom equal to (k-1) (n-1). Tlie

value, y. . is the deviation of the item from the model, or item misfit,ij

and is determined by taking the difference between the observed and

expected frequency of the examinees at a given ability level who

answered a given item correctly. Tliis difference was then divided

by the standard deviation of the observed frequency, squared and

summed over items and score groups. The BICAL program standardizes

these deviations (y- .) in computing the mean square fit statistic;

therefore, y. . has a normal distribution with a mean of zero and

standard deviation of one (Hambleton et al., 1977). Items with large

mean square fit values are items which do not fit the model. As shovm

in Table 7, the mean and standard deviation of the mean square fit

statistic increased with sample size.

The item difficulty estimates based on the Rasch model also have

an expected mean of zero and standard deviation of one (Wright and

Mead, 1975). These estimates remained very similar across sample

size and exceptionally close to the expected values (Table 7).

The Rasch model does not provide a parameter for item discriminating

power as all item discriminations are considered equal and centered

at one (Wright and Mean, 1975). Tlie BICAL program provided, as part

of the normal output, estimates of the item's discriminating power

to check the fit of the data to the model. The item discriminations

were obtained by regressing the difficulty of the item for each ability

group on the ability estimate of the group (Wright and Mead, 1975, p. 11).

65

Tlie means and standard deviations for the item discrimination estimates

were shown in Table 8 for each sample size.

TABLE 8

DESCRIPTIVE DATA ON ITEM DISCRIMINATION ESTIMATESBASED ON THE RASCH MODEL ACCORDING

TO SAMPLE SIZE

N = 250^

K = 53bN = 500K = 35

Mean

StandardDeviation

1.03

.28

1.02

.19

N = sample size

K = number ot items

N = 995

K = 33

1.03

.22

From the data in Table 8, the mean item discrimination estimates

appear nearly equal for each sample size, and quite close to the mean

expected value of one.

The best 15 and 30 items were then selected by each item analytic

procedure based on the information in Tables 5-7, and have been listed

in Tables 9 and 10 respectively.

The items selected under classical item analysis were determined by

the magnitude of the biserial correlation, e.g., the 15 and 30 items

having the higliest biserial correlations with total test score were

selected. Indices of item difficulty have been reported for inspection,

but in no way influenced the selection of items for classical item

analysis.

66

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The selection of items under factor ai^alysis was determined by

the item loadings on the first unrotated principal component. The

15 and 30 items having the highest item-component biserial

correlation were selected.

The selection of the 15 and 30 items from the Rasch analysis was

determined by the mean square fit of the item to tlie Rasch model. The

closer the mean square fit was to zero the better the item fit the

model, thus items with tl\e lowest mean square fit statistic were

selected.

Double Cross-Validation

After the tests of the best 15 and 30 items were developed by each

procedure, they were scored on independent samples, in a double cross-

validation procedure as noted in Table 3, Chapter III. Item and

test statistics, needed to test the five hypotheses were obtained for

the 15 and 30 item tests based on the cross-validation samples using the

GITAP program (Baker and Martin, 1970).

The GITAP program provided the following output:

each subject's total test score

test mean and standard deviation

. internal consistency estimates as measured by Hoyt's

analysis of variance procedure

. estimates of the standard ei'ror of measurement

. indices of item difficulty and biserial correlations

Comparison of the 15 Item Tests on Precision

The descriptive statistics based on the double cross-validation

samples for the 15 item tests have been presented in Table 11.

70

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71

The values of the iiiternal c^^isistency estimates for the tests

developed using the Rasch model were consistently lower than the

internal consistency estimates of the tests developed by classical

item analysis and factor analysis across all sample sizes.

Tlie observed internal consistency estimates were tested for

significance using confidence intervals described by Feldt (1965), to

see if they were statistically different from the internal consistency

estimate for the projected population using the Spearman-Brown Prophecy

Formula.

The internal consistency estimate for the population based on the

original 50 item subtest was .88 (Table 1). By applying the

Spearman-Brown Prophecy Formula (Mehrens and Lehman, 1975) the projected

population internal consistency estimate for a 15 item test was found

to be .687. The value .687 was the expected internal consistency if 55

of the 50 items were randomly deleted. Thus, confidence intervals

were generated around the observed internal consistency estimates,

presented in Table 11, for each procedure across all sample sizes to

see if any of the three item analytic teclmiques would produce a more

reliable test than would be expected from mere random item deletion.

The confidence intervals for the observed consistency estimates

for each procedure have been reported in Table 12.

When tlie sample sizes were 250 and 995 eacli item, analytic technique

produced an internal consistency estimate that was significantly different

from the projected population estimate (.687) at a confidence level of

95 percent. Each of the tl;ree techniques systematically retained

the 15 most homogeneous items. These tests were more precise

in terms of internal consistencv than would have been found if the

72

items were randoml)- deleted as noted by comparisons to tlie projected

population reliability coefficient.

Table 12

CONFIDENCE INTERVALS^ FOR THE OBSERVED INTERNALCONSISTENCY ESTIM/\TES BASED ON THE 15

ITEM TESTS ACCORD I F^G TO SAMPLE SIZE

95 V Confidence Interval

Procedure N = 250 N = 500 N = 995

Classical .748 - .828* .792 - .838* .810 - .841

Factor Analysis .760 - .837* .786 - .854* .797 - .831

Kasch .704 - .799* .688 - .757 .711 - .759

*

*

*

The £ values used in calculating the confidence intervals were obtainedfrom f'larisculo (1971).

+

Statistical significance is indicated when tlie population internalconsistency estimate is not concluded in tlie confidence intervalgenerated for each observed internal consistency estimate. Theprojected population value was .687.

Only two procedures produced tests with internal consistency

estimates significantly different from the projected population estimate

when tlie sample size was 500, classical item analysis and factor

analysis

.

As sample size decreased, in most cases, the internal consistency

for each method tended to decrease (Table 11). An exception was noted

for the Rasch tests, when the sample size decreased from 500 to 250,

internal consistency improved slightly.

The data reported in Table 11 indicated that the standard error

of measurement for the 15 item tests based on the Rasch model were

75

consistently larger than the stanu.ird error of nicasurcincnt of the

tests developed from classical item analysis and factor analysis for each

sample size. However, these differences were not meaningful in that

the difference did not equal or exceed 1.00 for any of the three

procedures.

The differences in mean item difficulties and discriminations

were tested for statistical significance to determine whether there were

differences in the types of items retained by eacli item anal}-tic method.

In tliis study item discriminations were measured by biserial correlations.

A two-way analysis of variance (fixed effects model) was performed

separately for the two dependent variables of item difficulty and item

discrimination. A check was made on the assumptions for the analysis

of variance to ensure that they were met. In tliese analyses, item

analytic technique and sample size were the two independent factors, each

with three levels.

For item difficulty, no significant differences were found for

item analytic technique, sample size, or their interaction, F (2,1261 =

2.57, p > .05; F (2,126) = .45, £ > .05; £ (4,126) = .33, £ > .05

respectively.

The means, standard deviations, and ranges of the item difficulties

based upon the 15 item tests have been reported in Table 13.

For the analysis of variance performed on the transformed

biserial correlations a significant F. ratio was observed for the factor

of item analytic technique, F (2,126) = 14.862, p < .05. No significant

differences were observed for sample size or the interaction of sample

size and item analytic technique for the transformed biserial

74

correlations F (2,126) - .30, p > .05; F C4,126) = 1.16, £ > .05

respectively. The means, standard deviations, and ranges of the

transformed biserial correlations, based upon the 15 item tests have

been presented in Table 14.

TABLE 13

15 ITEM TESTS:DESCRIPTIVE STATISTICS FOR ITEM DIFFICULTY

BY PROCEDURE AND SAMPLE SIZE

75

between tlic mean item discriminations liave been reported in Table 15,

TABLE 14

15 ITEM lESTS:

DESCRIPTIVE STATISTICS FOR ITEM DISCRIMINATIONS'

BY PROCEDURE AND SAMPLE SIZE

Mean

Procedure

Classical

FactorAnalysis

55.60 54.49

Rasch

43.51

Standard

76

TABLE 15

POST HOC COMPARISONS OF THE DIFFERENCES

BETWEEN THE MEAN ITEM DISCRIMINATIONS^FOR 11 IE 15 ITEM TESTS

Means

54.49 53760 43.51

Factor Analysis .889 10.98**

(54.49)

Classical ItemAnalysis 10.09**

(53.60)

Rasch Analysis(43.51)

^Based on transformed biserial correlations. Tlie transformation was a

linear transformation of the Fisher z_ statistic and multiplication

of the constant 60.241 providing a range of 0-100 for the biserial

correlation (Davis, 184b).

**£ < .01, HSD =6.88.

By increasing the test length to 30 items, the internal consistency

estimate was increased across each method and sample size, but a pattern

similar to that for the 15 item test emerged. The internal consistency

estimates from the test based on the Rasch model were slightly lower

than the internal consistency estimates for the tests based on

classical item analysis and factor analysis. The observed internal

consistency estimates were tested for significance, using the confidence

intervals described in the previous section, to see if they were

statistically different from the internal consistency estimate for the

population.

The projected population internal consistency estimate for a 30

item test was found to be .814 (via the Spearman-Brown Prophecy Formula).

77

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The value of .814 indicated tlio expected internal consistency

if 20 of the 50 items were randomly deleted.

Based on tlie observed internal consistency estimates reported

in Table 16, confidence intervals were generated for each item analytic

procedure and have been presented in Table 17.

TABLE 17

CONFIDENCE INTERVALS^ FOR THE OBSERVED INTERNAL

CONSISTENCY ESTIMATES BASED ON THE 30

ITEM TESTS ACCORDING TO S.AMPLE SIZE

95"b Confidence Interval

N = 250 N =500 N = 955

Classical .800-. 865 .845-. 880* .850-. 874*

Factor Analysis .825-. 881* .834-. 871* .848-. 873*

Rasch .794-. 860 .817-. 858* .838-. 863*

Tlie ¥_ values used in calculating the confidence intervals were obtainedfrom~Marisculo (1971).

*

Statistical significance is observed when the population internalconsistency estimate is not included in the confidence intervalgenerated for each observed internal consistency estimate. Tlie

projected population value was .814.

For the sample of 250 examinees, only one item analytic technique

(factor analysis) produced an internal consistency estimate that was

statistically different from the projected population estimate at a

confidence level of 95 percent.

However, all three techniques produced tests with internal con-

sistency estimates significantly different from the projected population

•

79

estimate when the sample was increased to 500, and 995. Thus, when

the number of examinees was large, each of the three teclmiques

procedures tests with higher internal consistency estimates than if the

test were produced by randomly deleting items.

For the 30 item tests, the effect of decreasing the sample size

tended to decrease internal consistency for each method (Table 16)

.

But the decrease was very slight.

The standard error of measurement was essentially tlie same for

the three methods of item analysis across the varying sample sizes.

Two-way analyses of variance were run on item difficulties and

item discriminations for the 30 item tests, similar to those run for

the 15 item tests. Again, the independent variables were item analytic

technique and sample size, each containing tliree levels.

No significant differences were observed for item difficulty

for the independent variables of item analytic technique, sam.ple size,

or their interaction, F. (2,261) = .46, p > .05; l_ (2,261 = .27, p > .05;

F_ (4,261) = .24, p > .05 respectively.

No significant differences were observed for the transformed

biserial correlations for the independent variables of item analytic

technqiue, sample size, or their interaction, F. (2,261) = 1.97, £ > .05;

F (2,261) = .74, £ > .05; F_ (4,261) = .48, £ > .05 respectively.

When the actual biserial correlations were tested in the two-wayanalysis of variance design similar F values were observed.

SI)

The means, standard deviations, and ranges of tlie item difficulties

and transformed biserial correlations based upon the 30 item tests

have been presented m Tables 18 and 19 respectively.

TABLE 18

30 ITBl TESTS: DESCRIPTIVE STATISTICS FOR ITEM

DIEFICULTY BY PROCEDURE AND SAMPLE SIZE

SI

Comparison of the 3U Item Tests on nfficiency

Lord (1974a, 1974b) proposed the formula used for approximating

the relative efficiency for tvv-o tests, stated previously in equation

15 as:

V . xCn - xj f

R.E. Cy,x) = -^^- ^ ^ '

X y(ny - y)f^

where R.E. (y,x) denotes the relative efficiency of y compared to x,

n and n are tlie numbers of items in the two tests, x and y are the

2'^

number-right scores having the same percentile rank, and f and f are' X y

the squared observed frequencies of x and y obtained from frequency

distributions for similar groups of examinees. A careful examination

of the formula for relative efficiency indicated tliat when n = n andX y

X = y, that it was the number of examinees at the specified ability

2 2level (f and f ) that determined the efficiency of the test. That is,

X y ' '

the fewer examinees observed at a particular percentile rank, the better

the test discriminates at that percentile rank. Therefore, test

efficiency was equated with the level of discrimination the test

was able to make between examinees, at various scores or percentile

ranks.

Three relative efficiency comparisons were made using the 30 item

tests based on the sample of 995 examinees. The three comparisons were:

[a) tlie test developed from factor analysis was compared to the test

developed by classical item analysis (b) the test developed by Rasch

analysis was compared to the test developed by classical item analysis,

and (c) the test developed from the Rascli analysis was compared to the

factor analytically developed test.

Hie efficiency curves for the thi-ce comparisons were shown in

Figure 2. The relative efficiency value was plotted on the ordinate,

while the percentile rank (student ability level] was plotted

along the abscissa. Computed values for the relative efficiency

comparisons liave been reported in Appendix B. A relative efficiency

of 1.00 would indicate that the tests are equally efficient.

T\\e test developed by factor analysis was more efficient for the

lower tenth of the pupils when compared to the test developed from

classical item analysis. Both the tests were about equally efficient

for the middle ability groups and high ability groups,

llie Rasch developed test was more efficient than the test based

on classical item analysis for average to high ability students

(40th-90th percentile rank). However, it was less efficient than the

classical item analysis test for students with very low or very high

abilities (lst-20tl"i percentile rank and 98th percentile rank].

When compared to the factorial ly developed test, the Rasch test

was again more efficient for students of average to high abilities

(50th-90th percentile rank] . The factorially developed test appeared

more efficient for the very low and very high ability students

(lst-20th percentile rank and 98th percentile rank).

Suimnary

Tlie results reported in this chapter are suiranarized for each of

the five hypotheses.

Hypothesis 1 . There are no significant differences in the

internal consistency estimates of the tests produced by the tliree

methods, as the nimiber of items decreases, when compared to the

83

20 30 40 50 60 70 80

PERCENTILE RANKS

1^

SO 99

key;

84

projected internal consistency estimates for the population for tests

of similar length.

Confidence intervals were calculated to test for differences between

the observed internal consistency estimates and the internal consistency

estimate for the population. As reported in Tables 12 and 17, for the

15 and 50 item tests, 15 of the 18 confidence intervals (at the 95 percent

level) generated around the sample estimate did not contain the population

value. This means that 15 of the observed internal consistency estimates

were superior to the population values projected for subtests of similar

length created by random deletion of items. Therefore, hypothesis one

was not supported. The procedures that produced the three observed

internal consistency estimates that were not significantly different

from the population value, and hence no different than would be expected

by random item deletion, were the Rasch procedure (15 item test, N = 500;

50 item test, N = 250) and the classical item analysis procedure (50

item test, N = 250)

.

Hypothesis 2 . There are no differences in the internal consistency

estimates of the tests produced by the three methods when the number

of examinees is decreased.

Hypothesis two was supported for the 15 and 50 item tests. Slight

decreases in internal consistency estimates were noted for the 15

item test (Table 11) as sample size decreased, but only decreases of

one or two one-hundreths of a point. Even smaller decreases were

observed on the 50 item test (Table 16)

.

H>T3othe5is 5 . There are no meaningful differences in the

magnitude of the standard error of measurement of the tests produced

by the three methods.

85

Hypothesis three was supported for the 15 and 30 item tests.

Meaningful differences were defined to be >_ 1.00, but none of the

three methods produced tests with standard errors of measurement that

differed by tliat much. In eacli case, the difference was approximately

one-tenth of a point or less (Tables 11 and 16)

.

Hy|:)othesis 4 . There are no differences in the difficulties or

discriminations of the items selected by the three methods.

Hypothesis four was supported for the 15 and 30 item tests with

respect to item difficulty. That is, the two-way analysis of variance

revealed no significant differences for either the 15 or 30 item tests

with regard to item difficulty.

Hypothesis four was also supported for item discrimination, but

only for the 30 item tests. Tlie two-way analysis of variance for item

discrimination indicated no significant differences for the 30 item

tests; however, on the 15 item tests, a significant £ ratio (p^ < .05)

for item analytic procedure was observed for item discrimination.

Tukey's HSD test revealed tliat items selected by the Rasch procedure

had significantly lower average biserial correlations than the items

selected by factor analysis and classical item analysis (Table 15)

.

This could have been expected because the range of the biserial

correlation was restricted when the items were originally selected

for the Rasch model. This procedure was necessary to meet one of the

assumptions for the Rasch model.

Hypothesis 5 . There are no differences across ability levels in

the efficiency of the tests produced by the three methods.

86

H)'pothesis five was not supported. The efficiency curves

illustrated in Figure 2, generally indicated tliat the tests based on

classical test theory were more effective for measuring students with

very low ability (20th percentile rank or less) and students with very

high abilities (98th percentile rank), llie Rasch developed test was

most efficient for assessing average and high ability students (40th-

90th percentile rank).

CHAPTER V

DISCUSSION AND CONCLUSIONS

This study was conducted to determine which of the three item

analytic procedures (classical item analysis, factor analysis, and the

Rasch model) miglu produce the superior test in terms of the precision

and the efficiency of measurement. A common item and examinee population

was used to test five hypotheses. Of the five hypotheses, three dealt

with elements of test precision as measured by internal consistency

estimates. Another hyjriothesis treated the issue of item discriminations.

Thus, it too was related to internal consistency. The fifth h>qiothesis

focused on the relative efficiency of the tests produced by tliree item

analytic teclniitjues. Tliis hy]oothesis altered the emphasis of the study

from one overall specific measure of a test's accuracy, in terms of

internal consistency, to a general comparison of each method as a

function of ability level. The discussion of the results then has been

focused in two major areas: (a) the precision of the tests, and [b)

the efficiency of the tests produced by the three methods of item analysis.

The Precision of the Tests Produced by the

Three Methods of Item Analysis

Each of the three item analytic techniques was applied to an in-

dependent sample to select the best 15 and 30 items. The stability of

the summary statistics across each sample size for the three item analytic

techniques indicated a tendency for tlie nine samples to be very homogeneous,

87

88

The similarity of the means, standard deviations, and percentages of

variance accounted for were noted on Tables 5-8, witli tlie exception

of the mean sciuare fit statistic (Table 7) whicli increased with

sample size. (Tlvis exception is discussed later in this cliapter.)

From these samples, items were selected by each item analytic techniciue

to maximize internal consistency.

The data reported in Tables 11 and 16 indicated the effectiveness

of each item analytic technique in producing internally consistent

tests. Before an overall decision can be made as to the superiority of

one technique over another, each of the h)q30theses relating to precision

must be considered.

Internal Consistency

Data in Tables 11 and 16 indicate that tlie two tests based on

classical test tlieory (factor analysis and classical item analysis)

appeared superior in terms of internal consistency wlien compared to the

tests developed by tlie Rasch model.

To test whether any of the three methods produced tests with

greater intei'nal consistency than a test created by random item deletion,

the internal consistency estimates were compared to the projected internal

consistency value for the population by using confidence intervals as

suggested by Feldt (1965). In order for a given sample internal consis-

tency estimate to be significant, the population value could not be

included in the confidence interval generated around that sample value.

For the 15 item tests, nine confidence intervals were calculated for

the nine estimates of internal consistency, one for each method at each

sample size. Eight of the nine sample values were shown to be significantly

greater than the population estimate at the 95 percent confidence level

(Table 12). Only tlie internal consistency estimate of the Rascli test.

80

based on the sample of 500 examinees, failed to reach a level significantly

greater than would have been expected by chance.

For the 30 item tests, nine confidence intervals were also calculated

for the nine estimates of internal consistency, one for eacli method

at each sample size. Seven of the nine sample internal consistency

estimates were shown to be significantly greater than the population

estimate at the 95 percent confidence level (Table 17) . The tests

based on classical item analysis and Rasch analysis, for the sample of

250 examinees, were not significantly different from the projected

population value for a 30 item test created by random item selection.

Therefore, for smaller samples (N = 250] factor analysis appeared

to be superior to classical item analysis and the Rasch analysis in

producing the most precise test.

Generally, as tlie number of examinees decreased so did the internal

consistency estimates. However, the tests based on factor analysis were

least affected by decreasing the sample sizes used in the cross-validation

for the 15 and 30 item tests (Tables 11 and 16).

Standard Error of Measurement

The standard error of measurement is the standard deviation of the

distribution of errors surrounding an individual's observed score on

an infinite number of parallel tests. Hence the smaller the standard

error of measurement, the greater the precision of the measurement. This

statistic is often considered a more meaningful measure of an instrument's

reliability than tlie reliability coefficient itself (Magnusson, 1966,

p. 82). Based on the data for this study, the standard errors of

measurement were consistently smaller for both the 15 and 30 item tests

90

produced by classical test theory as compared to the 15 and 30 item

tests based on the Rasch model; however, the differences in the standard

errors of measurement did not equal or exceed 1.00 for any of the

methods.

Types of Items Retained

Item difficulty . Item difficulties of the 15 and 50 item tests

were analyzed in separate two-way analyses of variance. The two

independent variables were sample size and item analytic technique.

No significant £ ratios were observed for eiti\er the 15 or 30 item

tests on item difficulty. Therefore, each item analytic technique

tended to select items whicli had similar item difficulties on the

average.

Item discrimination . In this study, the item discriminations were

measured by biserial correlation. Transformed biserial correlations

for the 15 and 30 item tests were analyzed in separate two-way analyses

of variance. The two independent variables were sample size and item

analytic technique. For the 15 item tests, a significant F ratio

(£ < .05) was observed for the main effect of item analytic technique.

The mean biserial correlation for each 15 i tem test s were 44 for the

Rasch test, 54 for the classical item analysis test, and 54 for the

factorial ly developed test. Tukey's USD post hoc comparison

indicated that the items selected by the Rasch procedure had lower

biserial correlations, on the average, than items selected on the basis

The actual mean biserial correlations for the three testscorresponding to the transformed biserial correlations were:.62, .71, ,71 respectively.

91

of factor analysis or classical i t om analysis. It should be noted

that this finding was due to the fact that the range of tlie biserial

correlations was restricted to .33 - .79 on the items selected for

the Rasch calibration. This was necessary to meet the assumption

of equal item discriminations.

However, when the test length was increased to 50 items, no

significant ¥_ ratios were observed for the variable of item discrimina-

tion. The difference in these two findings for the 15 and 50 item

tests can be explained by the way the 15 and 50 item tests were constructed.

The 15 item test was made up of the 15 items with the highest biserial

correlations. The 50 item test was made up of the above 15 items and

an additional set of 15 items with the next highest biserial correlations.

The addition of 15 more items meant that their average biserial correla-

tion was something less than the original 15 items. Hence, the mean

biserial correlations were reduced for the longer 50 item tests

[Table 19)

.

Conclusions

From the data presented for each of the four areas above, it was

concluded that each of the three item analytic techniques tended to

produce tests that were really no different in terms of the precision

of measurement.

Thus, the question to consider now is: Should practitioners

in the field of measurement spend their time learning to use the Rasch

12model to develop cognitive norm-referenced tests knowing the extra

11. . r

"Tlie field of test development is limited to cognitive norm-retcr-

enced tests because tliat was the type of instrument used in this study.

92

work and sopliisticatioii of knowledp,e required to effectively use the

Rasch procedures? Witli the criterion of internal consistency as a

measure of test superiority, it appeared from this study that time

spent factorial ly developing tests, or if computer facilities were not

available, tlie use of classical item analysis procedures seem more

than adequate for good test construction.

However, it must be remembered that internal consistency may

not be a fair and sufficient criterion. Internal consistency is an

integral part of classical test theory and may be biased since it was

derived from the classical model. Following tliat reasoning, Whitely

and Dawis (1974) liave commented on the precision of tests developed

using classical item analysis and Rasch analysis. They stated that if

the goal of item selection was to develop fixed-content tests, then

the classical teclmiques of item selection will yield the more precise

test since precision is specific to tlie trait distribution in a given

test. Whitely and Dawis indicated that tlie strength of the Rasch

analysis was in the individualized selection of items, as in tailored

testing, rather tlian the construction of fixed-content tests.

In considering the above situation, Lord (1974b) has stated that

internal consistency is an overall estimate of a test's homogeneity,

but provides no information on how the test as a whole discriminates

for the various ability groups taking tlie test. Thus, the three

techniques of test development were compared using an additional criterion,

relative efficiency.

93

Tlic F.ffjcicncy of ; Tests Produced by the

Tliree Methods of Item Analysis

The review of the literature concerning the relative efficiency

of a test presented in Chapter 1I-, cited studies mainly dealing with

latent trait theory (Birnbaum, 1968; Hambleton and Traub, 1971, 1973).

Studies comparing the relative efficiency of tests developed by latent

trait theory to classical test theory appear to be missing from the

literature on test efficiency.

The relative efficiency formula (Lord, 1974a, 1974b) was not

derived for any specific test development theory; therefore, relative

efficiency estimates should be applicable to any test development

technique. Lord (1974b) suggested his formula may not work well for

extremely short tests and that it should only be used on large samples

of examinees. Thus, in the present study, only the 30 item tests were

compared using the sample of 995 examinees. The three comparisons of

relative efficiency were: (a) the test based on factor analysis was

compared to the test based on classical item analysis, (b) tlie test

based on the Rasch analysis was compared to the test based on classical

item analysis, and (c) the test based on the Rasch analysis was compared

to the test based on factor analysis.

Generally, the results indicated that the Rasch test was superior

to the two tests based on classical test theory for students of average

and high ability. The two tests based on classical test theory, how-

ever, were superior in efficiency to the Rasch developed test for very

low and very high ability students (Figure 2). Test efficiency has been

defined as a measure of how well a test is able to discriminate between

examinees of varying abilities. Therefore, the test constructor must

94

ask himself, for which segment (s) of the examinee population is the

test intended to discriminate?

In this study, tlie test under consideration was a verbal aptitude

college admissions test. Usually college admissions officers are

interested in selecting students who will be successful once admitted

to college. The examinees wtio score very high on college admissions

tests will generally be admitted to college witliout any question. Thus,

it is less important to be able to discriminate among the very high

scoring examinees tlian to discriminate among the students who score near

the mean or in the upper middle range on a college admissions test.

For these students it is difficult to decide who sliould be admitted

and who sliould be denied admittance. If it is known that the admissions

test discriminates very well for average to high ability students, then

tlie reliability of the selection process based on test scores should

be increased. The data in this study, therefore, indicate tliat the

test based on the Rasch analysis would be most efficient for selecting

the average to 'nigh ability students for admission to college.

Lord (196S) has illustrated a very important feature of test

information and relative efficiency curves. He has shown tliat the

contribution of each item to a test is independent of all other items.

Thus, when information curves are available on a pool of items they can

be added to a test to achieve a prespecified information or relative

efficiency curve for any subpopulation of examinees (Lord, 1968)

.

Therefore, by using measures such as relative efficiency and information

curves psychometricians are able to develop very discriminating tests

for any segment of the population.

9 5

Conclusions

It has been suggested that because efficiency and test information

curves are a function of ability, tliese estimates ought to replace

the use of classical reliability estimates and the standard error of

measurement in test score information (Hambleton et al., 1977). This

suggestion certainly deserves some consideration in light of the present

study where it was shown that the three methods of item analysis

produced similar tests in terms of precision, but the three methods

produced very different tests in teimis of efficiency. Today, with

the increasing use of computers in test construction, perhaps the

more meaningful question to be asked by psychometricians is; For

which ability group is the test superior? Only test information curves

and measures of relative efficiency can answer that question.

Implications for Future Research

The results of this empirical study revealed that tests developed

using classical test theory, in spite of its inherent weaknesses, were

no different with respect to precision of measurement than tests

developed using one of the latent trait models, the one-parameter

Rasch model. Comparisons of relative efficiency for the 30 item tests

showed that the tests based on classical test theory were superior to

the Rasch developed test for very low and very high scoring examinees,

and the Rascli developed test was more efficient for average to high

scoring examinees on the verbal aptitude college admissions subtest

used in this study.

Only one of the four latent trait models, the Rasch model,

was used in this comparative study of test development techniques.

Perhaps it was the very nature of tliis simple model that resulted in the

96

devc lo])mcnt of ctjuivalcnt tests when compared to tlic tests developed

by classical item analysis and factor analysis in terms of precision.

It may be that the more technical- two- and three-parameter logistic

models would have produced tests comparable or superior to those

developed by classical test theory in terms of precision of measurement

and overall relative efficiency.

The two-parameter model allows for varying item discriminations so

that the initial selection of items would not have to have been restricted

to a prespecified range. The three-parameter model not only allows

for varying item discriminations, but also for the effects of guessing

on the test. It is reasonable to suspect that guessing may have been

a factor in the item scores for the t>T3e of cognitive test used in the

present study. Thus, before the findings of the study can be generally

accepted, replication is needed using not only other populations and

other instruments, but also other latent trait models.

If other latent trait models are to be considered in addition to

the Rascli model, several points need to be evaluated. The Rasch model

is the only latent trait model that provides for the direct calibration

of items and abilities based on unweighted "number right" scoring

(Wright, 1977). The two- and three-parameter models require a more

complex scoring system where the item response is weighted in order to

estimate the discrimination and guessing parameters. The weigliting

is an iterative process that may never converge or stablize unless

arbitrary boundaries are established [Wriglit, 1977, p. 104). Dccause

of tliis complex scoring system, the two- and thrcc-iniramcter logistic

models are less efficient for parameter estimation than the one-jiaram-

eter model in terms of computer time.

97

A further criticism of the two- and tiiree-parameter logistic models

has been offered by Wright (1977) concerning the additional item

parameters. If item discrimination parameters and item guessing

parameters are introduced into a theory of measurement, why not

person parameters for sensitivity to difficult items, and inclination

toward guessing? Wright questions wliether psychometricians really

need to make measurement theory so complex.

A final point to consider when comparing the latent trait models

is that only the one-parameter Rasch model provides a ratio scale

of measurement in terms of the calibrated item and ability scores

(Hambleton et al., 1977). Success on a particular item is given by

the product of the person's ability and the item's easiness. Tlius, the

person with no ability will have zero odds or probability of success

on any item. The same logic applies to items with no easiness (zero

difficulty) they cannot be solved. Thus, measurements made with Rasch

calibrated items are on a ratio scale; and it is the ratio scale of

measurement that leads to the concept of specific objectivity.

Tlierefore, it seems that each latent trait model has its own

set of advantages and disadvantages and sliould be considered if com-

parisons are to be made to the classical models for test development

purposes.

Additional areas for future research may lead to actual comparisons

of the content of the items selected by each of the three item analytic

techniques. Davis (1951) and Co.\ (1965) have criticized the selection

of items solely on statistical criteria. The use of statistical criteria

alone, may result in changing the nature of the trait being measured

by deleting the items essential to adequate coi;tent coverage. Whitely

98

and Dawis fioyb) found this to be true in a study of verbal analogy

items wliere the type of relationship and content of specific analogy

items proved to be quite significant when studied in isolation.

A restricting factor in this stud)' was tliat a prespecified

number of test items were selected, e.g., 15 and 50, by each item

analytic procedure. For the Rasch procedure, perhaps some number of

items less than 30, but greater than 15 may have provided a better fit

to the model. This procedure might have produced mean square fit

statistics that were equivalent across the sample sizes rather than

increase witli sample size as found in this study. Thus, selecting

the number of items precisely fitting the Rasch model and comparing

that number of items with classical item analysis and factor analysis

miglit liave led to different conclusions than those made in the present

study with regard to the precision and relative efficiency of measurement,

CMPTF.R VI

SlIffMARY

The quality of the items in a test determine its validity and

reliability. Tlirough the application of item analysis procedures, test

constructors are able to obtain quantitative objective information useful

in developing and judging the quality of a test and its items.

Classical test tlieory forms the basis for one method of test

development. An integral part of the development of tests based on the

classical model is the utilization of classical item analysis or factor

analysis. Classical item analysis is a procedure to obtain a description

of the statistical characteristics of each item in the test. This

approach requires identification of single items whicli provide maximum

discrimination between individuals on the latent trait being measured.

Theoretically, selecting items which have a high correlation with total

test score will result in a discriminating test which is homogeneous

with respect to the latent trait. Therefore, classical item analysis

is an aid to developing internally consistent tests.

An alternative metliod of test development, but based on the

classical model, is factor analysis. Factor analysis is a more complex

test development procedure than classical item analysis. It is a

statistical technicjue that takes into account the item correlation with

99

100

all other individual items in the test simultaneously. Groups of

similar items tend to cluster together and comprise the latent traits

(factors) underlying the test. Thus, under the classical model then,

classical item analysis can be viewed as a unidimensional basis for

item analysis, less sophisticated than the multidimensional procedure

of factor analysis.

Classical item analysis and factor analysis have long been the only

techniques described in measurement texts for use in test development

(Baker, 1977). However, with the publication of Lord and Novick's

Statistical Theories of Mental Test Scores,

(19b8) considerable attention

is being directed now toward the field of latent trait theory as a new

area in test development. Proponents of this approach claim that the

advantages of latent trait theory over classical test theory are

twofold: (a) theoretically it provides item parameters that are in-

variant across examinee samples which will differ with respect to the

latent trait, and (b) it provides item characteristic curves that give

insight into how specific items discriminate between students of

varying abilities.

Four latent trait models have been developed for use with

dichotomously scored data: The normal ogive, and the one-, two-, and

three-parameter logistic model (Hambleton and Cook, 1977; Lord and

Novick, 1968). This study was concerned with the one-parameter logistic

Rasch model because it is the simplest of the four models.

A review of tlie literature revealed numerous studies conducted

in each of the tliree areas of item analysis, but relatively few

comparative studies were reported between the three methods. Missing

101

from the review were comparative studies among all three, item analytic

techniques. Therefore, the present study was designed to compare the

methods of classical item analysis, factor analysis, and the Rasch model

on measures of precision and relative efficiency in test development.

An empirical study was designed to compare the effects of the three

methods of item analysis on test development across different sample

sizes. Item response data were obtained from a sample of 5,235 high

school seniors on a cognitive test of verbal aptitude.

The subjects were divided into 9 samples: three independent

groups of 250 subjects each, three independent groups of 500 subjects

each, and three independent groups of 995 subjects eacli. The independent

groups were obtained so that tests of statistical significance could

be performed.

The item response data were then analyzed in three phases. First,

the "best" 15 and 30 items were selected using each item analytic

technique. Under classical item analysis, the best 15 and 30 items

were selected based on the highest biserial correlations. For factor

analysis, the best 15 and 30 items were selected based on the highest

item loadings on the first (unrotated) principal component. The

selections of the best 15 and 30 items using tlie Rasch model were

based upon the mean square fit of the items to the model. These

procedures were used for each group of subjects. Second, a double

cross-validation design was employed to obtain estimates on the item and

test parameters for the best 15 and 30 items. Tlie items selected from

tlic three item analytic teclmiques were scored for different samples

of subjects by randomly reassigning the samples wliich Iiad been used

in the original item analysis.

102

Then, the best 15 and 30 itc.is chosen by each method were submitted

to a common item analytic procedure in order to obtain estimates for

comparing the three item analytic methods. Third, a two-way analysis

of variance and a Tukey HSD post hoc comparison test, when indicated,

were used to test for differences in the properties of items selected

by each item analytic procedure. Also confidence intervals were

calculated to compare the internal consistency estimates to a population

value. In addition, the relative efficiencies of the 30 item tests

developed by each item analytic technique were compared for the sample

of 995 subjects.

The results of the analysis showed that there were no apparent

differences in the types of tests produced by the three methods of item

analysis in terms of the precision of measurement. The three methods were

compared on measures of internal consistency, the standard error of

measurement, mean item difficulty, and mean item discrimination.

Confidence intervals were generated around the observed internal

consistency estimates for the 15 and 30 item tests produced by each

method. The confidence intervals were obtained to compare the observed

internal consistency estimates from each test to the project population

internal consistency for tests of a similar length as suggested in

Feldt (1965) . The projected population value (obtained via Spearman-

Brown Prophecy Formula) represented what the test's internal consistency

would have been for a test created by deleting items at random. Of the

18 confidence intervals calculated at a 95 percent level of confidence,

15 did not contain tlie projected population value. Therefore, it was

concluded that the three item analytic techniques were significantly

different from random item deletion in producing tests with liigher

internal consistenc\' estimates. It was also noted that as the

103

number of examinees decreased so did the internal consistency

estimates.

The standard error of measurement was consistently smaller for

both the 15 and 30 item tests produced by classical test theory when

compared to tlie 15 and 50 item tests based on the Risch model. However,

the differences in the standard errors of measurement did not equal or

exceed 1.00 for any of the methods.

No significant F ratios were observed for either 15 of 30 item

tests on item difficulty. Therefore, each item analytic technique tended

to select items which had similar item difficulties on the average.

For the variable, item discrimination on the 15 item tests, a

significant £ ratio (p^ < .05) was observed for the main effect of the

item analytic technique. Tukey's USD post hoc analysis indicated that

tlie Rasch test tended to contain items with lower biserial correlations,

on the average, tlian tests procedured by factor analysis and classical

item analysis. This finding was probably due to the fact that the range

of the biserial correlations was restricted to .39 - .79 for items

retained in the Rasch analysis. However, when the test length was

increased to 30 items, no significant F_ ratios were observed for the

variable of item discrimination.

In terms of the test efficiency, the results indicated substantive

differences in tlie tests produced by the three methods of item analysis.

The 30 item test for the sample of 995 examinees was used in this com-

parison. It was found that the Rasch developed test was superior to

the two tests based on classical test theory for students of average to

high ability. The tests based on classical test theory, however, were

superior in efficiency to the Rasch test for students of very low

10

#

or very high ability. In light of these findings, it was suggested

that measures of test efficiency ouglit to be incorporated into the

test development procedures, as it provides much more detailed

information on how the test discriminates for various ability groups

than does a single overall estimate of a test's homogeneity.

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23-48.

APPl-NDIX A

MATHEMATICAL DERIVATION OF mE RASCH MODEL^

Summarized from Ryan (1977)

.NLVniEM-VriCAL DERIVATION OF lllE lUSCll MODEL

If ability is equal to and item difficulty is equal to i; then

the odds of correctly solving an item is given by

,^ e , (1)odas =:

^ '

s

Ulien 6>£, the person will get the item right, when 9<c the person will

get the item wrong, and when 9 = ^ the person will have a 50-50

chance for success on the item.

Tlie probability of a correct response can readily be derived from

the statement of the odds. In general,

oddsProbability = ' '''''

1 + odds

substituting (1) into (2)

P -•

C5)

1 + e/c

which is equivalent to

p . ^/^ = ^1^-.

^^ (4)

c/c + e/c c + Q/c c + e

Formula 4 is the prLbabiHty of a correct response.

Tlie formula of an incorrect response is represented by Q, which is

1-P. Thus,

Q = (1-P) = 1 -^

, (5)

; + 6

which is equivalent to

Q = c + 9 - 6 = c . (6)

c. + Q Q + i> C + 9

112

115

Formula 6 is the probability of an incorrect response.

Tlie relationship of these probability statements to the statement

of the odds is given by

Q ^/c + e c

The right side of equation 7 is simply the odds as defined in equation 1.

The left side of the equation is the probability of a correct response

divided by the probability of an incorrect response. The probability of

a correct response, P, is estimated by the proportion of examinees in

a sample who correctly answer an item. On a test of k items, the pro-

bability of a person with score of j (where 1 ^ j ^ 1<) , correctly

answering a particular item is simply the proportion of people with

the raw score j who correctly answered the item. This is nothing more

than the item difficulty for all the people who have a raw score of j.

The probability can be calculated in terms of the item difficulty for

all raw score groups from 1 to (k-1) and across all items.

The probability of an incorrect response, Q, is the proportion of

people answering an item incorrectly. Tliis is simply one minus the

proportion answering it correctly (1-P) . Both P and Q are easily

calculated from a set of data hence the value of P/Q in formula 7 is

an easily derived statistic which estimates the odds.

Separating the Parameters

Consider again equation 7 and take the natural log (In) of both

sides of the equation. This gives

In (^) = In (|) . (8)

\u

Since the In of a rtitio is the same as the difference of the In.

8 becomes

In (|) = In 6 - In C • (9)

Person Free Item Difficulties

Next consider a score group with ability 6 and two test items

onwith difficulties c and c, respectively. The probability of a pers

with ability 6 correctly answering an item of difficulty r is P.,.

The probability of an incorrect response is Q, , . Tlie probability of

a person with ability 6 correctly answering an item of difficulty ^

is P.-, and tlie probability of an incorrect response is Q -,. By

equation 9,

PllIn (^) = In 6 - In r and (10)

^11

In (^) = In 9 - In c.. (11)^12

If equation 11 is subtracted from equation 10, term by term, the result

is

P P

In (A - In {—) = (In 9 - In Cj - (In 6 - In ^) . (12)^11 ^12

Tills is the same as

P P

In (^) - In (^) = In 9^ - In c^ - In 9^ + In r,^, (13)

P P

or. In (^) - m (^) - In c^ " In C^- (14)

Ec[uation 14 should be examined very carefully. On the left side of

the equation is an easily calculated statistic: The difference

between the In odds of a correct response on item 1 compared to item 2.

115

The right side is significant for what it does not contain. There is

no parameter for tiie person's ability on the right side of tl\e equation.

Tliis same result occurs regardless of the ability of the person or

group examined. The difference between the difficulty of item 1 and

the difficulty of item 2 can be calculated independently of the

subject or group of subjects involved. In general, for any two items

of difficulty Ci and ; , the difference between the difficulty of the1 m _^. „ „ . „ f_^

two items is given by

in C„^ - in C^ ^ in iA - In (^) . (15)

1 1 i m

Item Free Person Abilities

The discussion of person ability estimates is an exact parallel

to the discussion of item difficulty estimates. Instead of comparing

two items across any group of examinees the discussion of ability

proceeds by comparing any two groups on any test item. Consider score

group with ability 6 score group ^ with ability Q^, and item with

difficulty Q . From equation 9,

^11In (^^—) = In 9 - In c,, and (16)

^11

^1In (—) = In e^ - In ^^ . (17)

Subtracting equation 17 from equation 16 will yield

P P

In i-A - In i-^) - In 8^ - In 6 (18)^11 ^21

The difference between the abilities of the examinees in score group ,

and score group ^ (the right side of equation 18) is described without

reference to the item involved. In general, for any two groups with

abilities 9 . and 9 .

,

1 J

lib

In 6

P., P..

In a = m (^) - In (^) (19)

for any item with difficulty c In this case the abilities are being

compared independently of the difficulty of the item used to compare

them. Tliis is often referred to as item free person ability estimation.

Formalizing the Model

To describe the Rasch model let ln6 and In C be re-defined.

Specifically, let,

e = In 6, and (20)

6 = In c. ^—.,._^^ (21)

Equations 20 and 21 simply define the In ability as B and the In

difficulty as 6.—

If both sides of equation 20 and 21 are raised to the base of the

natural log system, e, we get

3 In , ande = e

6 In c.e = e

Recall equation 5

P = 6/C

(22)

(23)

(24)1 + e/c

and substitute the equivalent terms for 9 and C as defined in equations

22 and 25. This gives

6

P =

P =

e / e or

1 + e / e

( 3-6 )

1 + e( 3-6

)

(25)

(26)

117

More formally this is

P (X^. = 1| 3,, 6.)ef \ - '0

l.e^h- ^^ (27)

Equation 27 is the Rasch model,

APPENDIX B

RELATIVP. EFFICIENCY VALUES USED IN FIGURE 2 FOR

THE COMPARISONS AMONG ITEM ANALYTIC METHODS

TABLE B.l

RELATIVE EFFICIENCY VALUES USED IN FIGURE 2 FOR

THE COMPARISONS AMONG ITEM ANALYTIC METHODS

BIOGRAPHICAL SKETCH

Iris Benson was born January 19, 1946, in Charleston, South

Carolina. She graduated from Hialeah High Scliool, Hialeah, Florida,

in June 1964.

She began attending college part-time in 1968, and graduated in

August 1971 from Santa Fc Junior College in Gainesville, Florida.- Iris

then attended the University of Florida, and received a Bachelor of

Arts degree with honors in March 1973 with a major in Psychology.

In the fall of 1973, she began her graduate studies in the College

of Education at the University of Florida. Upon completing her master's

thesis entitled. The Florida Twelfth Grade Testing Program: A factor

analysis of tl^e aptitude and achievement subtests , she received the

degree Master of Arts in Education in June 1975.

Iris began her doctoral program at the University of Florida in the

fall of 1975. Wliile working on \\ev graduate degrees slie has held, at

various times, a graduate assistantship in tlie area of evaluation and

test develo]Miient , and teaching assistantships in graduate level courses

in educational measurement and statistics. She was also an evaluation

intern at the Northwest Regional Educational Laboratory in Portland,

Oregon from September 1976 to March 1977.

Iris is a member of the American Educational Research Association,

and the National Council on Measurement in Education. She has co-authored

several articles and papers on the topics of examinee test-taking behavior,

appropriate statistics for different measurement scales, and the adversary

evaluation model.

<

120

I certify that I have read this study and that in my opinion itconforms to acceptable standards of scholarly presentation and is fullyadequate, in scope and quality, as a dissertation for the degree ofDoctor of Philosophy.

(•....,• : ,(riA

William B. Ware, ChairmanProfessor of Foundations of

Education

I certify that I have read this study and that in my opinion it

conforms to acceptable standards of scholarly presentation and is fullyadequate, in scope and quality, as a dissertation for the degree ofDoctor of Plulosophy.

Linda M. Crocker -

Associate Professor of Foundationsof Education

I certify that I have read this study and that in my opinion it

conforms to acceptable standards of scholarly presentation and is fullyadequate, in scope and quality, as a dissertation for the degree ofDoctor of Philosophy.

L•v^^^^ \\l.u-t^Ahn M. Newell

P]3ofessor of Foundations ofEducation

I certify that I have read this study and that in my opinion itconforms to acceptable standards of scholarly presentation and is fullyadequate, in scope and quality, as a dissertation for the degree ofDoctor of Philosophy.

William R. PowellProfessor of Instructional

Leadership and Support

This dissertation was submitted to the Graduate Faculty of theDepartment of Foundations of Education in the College of Education andto the Graduate Council, and was accepted as partial fulfillment of therequirements for the degree of Doctor of Philosophy.

December 1977

I01^'C tfKc<.-Chairman, Foundations ofEducation

Dean, Graduate School

^7

4^F^13 7 8. 03.8.5.